Chemical reaction networks and diffusion

Transcription

1 FYTN05 Computer Assignment 2 Chemical reaction networks and diffusion Supervisor: Adriaan Merlevede Office: K , adriaan@thep.lu.se 1 Introduction This exercise focuses on understanding and modelling the dynamical properties of chemical and biochemical reaction systems. This document contains a recapitulation of the relevant theory, and a set of questions which should be answered and presented in a cohesive report. We will start by modelling simple chemical reactions, and continue with enzymatic catalysis. We will also look at the effects of including a spatial dimension with discrete diffusion-reaction models. The assignment is accompanied by an ODE simulator, which is intended to illustrate the dynamical behaviour of the reaction and reaction-diffusion systems we look at. It will output the simulated data to a text file, which can be used for adding figures to the report. The simulater has a simple GUI and no Java programming is required for this course, but the jar file contains the source code for those who are interested. 2 Assignment description Your job is to write an introductory text explaining the theory of chemical reaction networks. The questions and simulations will guide you to gain some more insight in the course material, and to serve as examples, which should be presented as part of the report. The goal of this assignment format is to exercise scientific communication, and to integrate the questions into a meaningful whole. Your grade will depend on your ability to answer the questions but also fulfill those two goals. Focus on the following guidelines when writing your report: The report is written in English. Feedback on spelling and grammar is intended only to help some students improve, as these are not factors in the grading. 1

2 But your ability to communicate ideas efficiently will be taken into account. You may judge for yourself how best to organise the report. That means you are not required to use a standard abstract-introduction-methods-resultsdiscussion-conclusions format. But keep in mind that the goal is scientific writing. Make good use of titles, paragraphs and introductions to guide the reader. Visualize. Figures are the first thing people see and the last thing they remember. Try to be efficient with your visualization and to make plots that show the point you want to make as clearly as possible. You may use any software. Some suggestions are Matlab / Octave, python (matplotlib), R (ggplot2), and Mathematica. A short guide to gnuplot is included in the appendix, but its use is optional. Do not insert screenshots from the Java program. Visualization is part of the grading. Be concise. Make sure to say everything you want to say, but try not to use unnecessary words or repetition. This goes for text but also for figures: if you are using multiple figures close together (for example, plotting the output of the ODE simulator for different parameter values), perhaps there is a better way to visually get your point across, or to do it in less space. It is often shorter and clearer to combine the results of multiple simulations into one figure, or to plot measurements other than concentration-as-a-function-of-time. Creativity is encouraged. Extra analyses, observations, interesting graphics etc. can be included. Any work that is not originally yours, including sources such as web pages and other students, should be properly cited, or will be reported to the university disciplinary council. Hand in the report in PDF format. 3 Background This section is a short overview of the theory covered by this assignment. Consult the testbook textbook Philips et al. s Physical Biology of the Cell, or other sources of information, for more detailed explanations, which may be required to answer some of the assigned problems (Note: The sections below do not correspond perfectly to textbook chapters.) You may also contact the course assistant if you have further questions. 3.1 Thermodynamics in biochemistry 2 See also:

3 Chapter 5: Mechanical and Chemical Equilibrium in the Living Cell Chapter 6: Entropy Rules! Most biochemistry occurs inside a cell: at constant temperature, pressure and in aqueous solution. Thermodynamically, such systems are best described in terms of the Gibbs free energy (G). The free energy may only decrease over time in a closed system, and the system is in equilibrium if and only if G is at its minimum ( second law of thermodynamics). Each (bio)chemical reaction changes the free energy of the system. Specifically, during a reaction, the reagents are removed from the system and the products are added, along with their associated free energies. Thus, the change of free energy G is given by (at constant temperature and pressure): G = i µ i N i (1) where N i is the number of molecules of species i in the system 1, and µ i is the free energy associated with each molecular species i. The quantity µ i is called the chemical potential, and in general dependends on the temperature and concentration c i. This dependence can be decomposed in two terms: a constant term (µ 0 i ) equal to the potential at some reference concentration (c 0 i ), plus a concentration-dependent term µ i T S ( ) G N i = E,Nj i N i T,P,N j i (2) µ i = k B T ln (c i /c 0 i ) + µ 0 (ϵ i, T ) (3) Using the above equations, G may also be decomposed into concentration-independent ( G 0 ) and -dependent terms. At equilibrium, we have G = 0, and we can derive: [X i ] N i = exp ( G 0 /k B T ) eq i =: K eq (4) where [X i ] is the concentration of the molecular species X i. This expression (either the left or right hand side, since they are equal) defines the reaction constant K eq. 1 Thus, N i is the (possibly negative) change of the number of molecules when the reaction occurs, or, equivalently, the number of molecules i in the product, minus the number of molecules i in the reagents. 3

4 3.2 Mass-action reaction kinetics See also: Chapter 15: Rate Equations and Dynamics in the Cell The law of mass-action states that the reaction rate (dr/dt, the number of times a reaction occurs per time unit) is proportional to the product of the concentrations of the reagents. In other words, if we have a general reaction of the form aa + bb +... k zz + yy +... (5) then the reaction rate is given by dr dt = k [A]a [B] b... (6) Knowing the reaction rate is an invaluable tool for modelling the dynamic behavior of reaction systems, because it allows us to derive ordinary differential equations relating the molecular concentrations to their time derivatives. The rate of change dr of a species X due to a reaction r is N X. If X is involved in multiple reactions, dt their contributions can be added together to obtain the total rate of change of [X]. Note that for any reaction that involves [X], the reverse reaction does, too. The proportionality constant k in the law of mass action is called the reaction s rate constant. It can be shown (derivation not part of this course) that k = A exp ( G /k B T ) ; A = k B T/h (7) where A is called the Arrhenius factor, and h is Planck s constant. The quantity G is the activation energy of the reaction. The activation energy is the free energy of the most unstable reaction intermediate, compared to the free energy of the substrate, i.e. the energy required to perform the most difficult part of the reaction. The relation between the different free energies is depicted graphically in Figure Enzymatic reaction kinetics See also: Chapter 15: Rate Equations and Dynamics in the Cell Many biochemical reactions are catalysed by specialized proteins called enzymes. A catalyst is a chemical that does not participate in a reaction as a reagent or product, but forms temporary stabilizing interactions with the reaction intermediaries. This effectively lowers the activation energy and increases the reaction rate, as illustrated in Figure 1. Catalysts such as enzymes can speed up reaction rates by several orders of magnetude in this way. Most important biochemical reactions are catalysed by 4

5 very efficient enzymes, and occur at negligible rates when the enzyme is not present. So usually the kinetics of the non-catalysed reaction mechanism can be safely ignored when modelling the dynamic behavior of biochemical reaction mechanisms. In order for an enzymatic reaction to occur, the substrates should not only interact with each other, but also with the enzyme. Enzymes typically form an intermediary complex with the reagents, waiting for other reagents to bind, or for the reaction to progres. These enzyme-substrates are fairly stable, so that they are best modelled as a separate reaction step. For a single-substrate enzymatic reaction E P catalysed by an enzyme E, we have: S + E k + SE k cat S + P (8) k From this we can derive the Michaelis Menten equation for enzymatic reaction kinetics: dr dt = V [S] max K M + [S] (9) Figure 1: (left) Progression of the free energy during a chemical reaction. The substrates undergo several intermediary reaction steps before ending up on the other side of the reaction. The activation energy is determined by the most unstable intermediary state, as compared to the substrate state. (right) A catalyst such as an enzyme may stabilize the intermediary reaction states, reducing the energy peak and increasing the reaction rate. 3.4 Diffusion See also: Chapter 13: A Statistical View of Biological Dynamics 5

6 The mass action law and Michaelis Menten equation describe reaction kinetics in a spacially uniform system: they were derived above under the assumption that any two molecules have an equal probability of interacting. In reality, this assumption is often incorrect, especially when dealing with systems composed of different compartments, for example different cells in a tissue, or different organelles in a cell. In biology, these compartments are often separated by semi-permeable membranes, which allow free transport of some molecules but are impassible to others. To deal with whis fact, we can describe the reaction system separately in each compartment (providing a different variable for each compound in each cell), and add appropriate terms to the differential equations to deal with transport of molecules between cells. Essentially, if we have n compartments, we split each variable [X] into n variables [X 1 ], [X 2 ],..., [X n ], and model transport from cell i to cell j as a reaction X i X j. Cells have multiple modes of transport at their disposal, which describe different kinetics for the "transport reactions" in the system. The simplest mechanism of transport is diffusion, which occurs for molecules that can pass freely through the membrane, often through specialized channels. In a diffusive process, each molecule makes a random walk between neighboring cells. Overall, the transport of compound X from cell i to a neighboring cell j occurs at a rate proportional to [X i ]. The proportionality constant is called the diffusion constant D. At equilibrium, diffusion tends to equal concentrations everywhere. However, in combination with chemical reactions, diffusion can lead to interesting stable or moving spatial patterns. Many of the beautiful patterns that decorate animals and plants are the result of reaction-diffusion mechanisms guiding pigment distribution. Figure 2. Pigment distribution pattern on the skin of the giant pufferfish (Tetraodon mbu), one of the spatial patterns that can be reproduced in some simple reaction-diffusion systems. 4 Problems 0.a) Derive equation 4. 0.b) Connect the theories of thermodynamics and reaction kinetics by relating equation 4 to the law of mass action (equation 6). 6

7 4.1 Mass action reactions Consider a simple two-state reaction: A k k + B You can explore this reaction in the Java simulator, using the TwoStateReaction setting (Y 0 and Y 1 are A and B, respectively). 1.a) Write down the differential equations for the time evolution of [A] and [B]. 1.b) Give three expressions for the equilibrium constant K eq. Let µ 0 A = 8 kj/mol, µ0 B = 3 kj/mol, and G = 75 kj/mol. 1.c) Calculate G 0, K eq, k + and k. 1.d) Describe the equilibrium in terms of [A] and [B]. How does it depend on the activation barrier G? 1.e) Do the initial conditions matter? In what direction is the reaction moving? How does G affect the system s behavior?. 4.2 Enzymes Imagine that the forward reaction is catalyzed by an enzyme, lowering the activation energy to G E. 2.a) What is the new value of k +? Did k change? Did the equilibrium? Try to get a feel for how changes in the enzyme efficiency can change the reaction properties, and imagine how this might affect a cell. 2.b) Is this a reasonable model of enzymatic reaction kinetics? Under which circumstances can it work or break down, and why? Let us now explicitly include the enzyme in our model for the reaction kinetics. A naive way of doing this is as follows: A + E k B + E 7

8 We assume that the reverse reaction occurs at a negligible rate. You can simulate this system in the Java program as SimpleEnzymatic, where X 1 = [A] and X 2 = [B]. Start with [B] 0 = 0. 2.c) Describe the reaction rate and write down the differential equations for the time evolution of the concentrations [A], [B], [E]. 2.d) Is this a reasonable model of enzymatic reaction kinetics? Under which circumstances can it work or break down, and why? Hint: Let [A] or [E]. Should the reaction rate still increase proportionally? We will compare this naive approach with Michaelis Menten kinetics, where we explicitly adopt an intermediary enzyme-substrate complex ES. You can use Michaelis- Menten as the Java program s reaction setting for this. 2.e) Derive the Michaelis Menten equation (equation 9), by assuming that the first reaction is always in equilibrium in the time scale of the second step. 2.f) Is this a reasonable model of enzymatic reaction kinetics? Compare with the two approaches above. 4.3 Diffusion We will introduce diffusion of substances between cells ordered in a one-dimensional grid: cell i neighbors cells i 1 and i + 1. The bottom section of the Java program allows you to manipulate the number of cells, and the diffusion constants of the different compounds (assuming that each pair of neighboring cells has the same diffusion constants), as well as add noise to the initial conditions so each cell starts out in a slightly different state. 3.a) Going back to the two-state reaction, give the differential equations describing d[a]/dt and d[b]/dt in a cell i, but now including terms modelling concentration changes due to diffusion to and from the left and right neighbour cells. 3.b) Show that this is equation is the discrete equivalent of the diffusion equation. 3.c) Starting from a molecular random walk process between cells in one dimension, show that the equation corresponds to passive (entropic) transport between cell. 8

9 4.4 Brusselator The Brusselator is a reactive system, first studied in the 1970s in Brussels. It is of interest because it exhibits some interesting oscillatory behavior. The Brusselator can lead to different kinds of patterns for different input parameters, particularly when introduced in a reaction-diffusion system. The reactions used are A k 1 X 2X + Y k 2 3X B + X k 3 Y + C X k 4 D. We assume that the reverse reactions are negligible, and are only interested in X and Y. Assume that the concentrations of A and B are large enough that they can be considered constant in the relevant time frame. Use the Brusselator reaction setting in the Java simulator. 4.a) Write down the differential equations describing the time evolution of [X] and [Y ]. 4.b) Explore and understand the effects of the different parameters when there is only one cell. Can you get the system to oscillate? Change the time scale if necessary. 4.c) Increase the number of cells to 100 and randomize the initial concentrations of X and Y. What does diffusion do? Compare no diffusion, diffusion of one compound, and diffusion of both. (Remember that both your text and plots should illustrate an insightful observation. Don t describe or plot raw data without making a point.) 4.d) Can you make the system exhibit a non-trivial spatial or spatiotemporal pattern? This requires plotting the spatial dimension, which is not visible in the Java simulator output. Hint: Use parameter values k 1 A = 0.1, k 2 = 0.1, k 3 B = 0.2, and k 4 = 0.1, with small random deviations in the initial concentrations. Find diffusion values that do not lead to a uniform system, or a set of independent cells. 9