Lab 12. Biochemical/genetic network modeling

Purpose of this lab

Use PLAS to build and explore the behavior of a simple biochemical or genetic pathway. PLAS (Power Law Analysis and Simulation) accepts your system of differential equations and simulates the behavior of a biochemical/genetic system operating by these equations.
Questions in this color should be addressed in your report.

Introduction

You'll need some but not much mathematics, so don't get worried. While an ordinary algebraic equation relates two or more quantities (variables), in a differential equation at least one of the terms is the rate of change or derivative of a variable with respect to another variable such as time. The variables we work with in biochemical modeling are quantities of a reactant molecule, such as a substrate, enzyme, cofactor or product. If we include genes in the model, we can think of the "quantity" of a gene as the amount of its protein product (say, a transcription factor, kinase enzyme, or other activating molecule) present in the cell . In PLAS, the quantity might be named X1, and then its rate of change per unit time would be written as X1'. The rules that we will construct to describe a network are of this form:

The rate of change in the amount (= concentration) of molecule X in the system at time point t is a function of the amounts of molecules X, Y, and Z at that time point.

For example, we might write

X1' = a Y - b Z - c X1

where a and b are constants to which we assign some numerical value such as 0.5 or 1e-4. In PLAS, the space between a and Y means the same thing as a multiplication symbol such as *.

PLAS will read all your rules, then compute the values of each of the variables (reactant concentrations) at each time point you specify. Finally it will plot a time course of these values. You may also ask PLAS to plot a phase plot, which plots one variable against another instead of against time.

First we'll learn the simple rules for constructing a system, then we'll play with the PLAS program.

If the PLAS 1.2 program has not already been installed in the course directory on your computer, download and install it.
Start PLAS and open Help/Contents. Read the Introduction, Overview, and Syntax reference.
Choose File/Open and open file Template.plc. You'll modify this and save it under another name. But to see what it does, choose Analysis/Run.
We will model a negative-feedback gene regulatory network for a generalized circadian clock, looking like the figure at right. The lines ending in arrowheads represent promotion (a positive rate of change) and those ending in crossbars represent inhibition (a negative rate of change). Of course, we don't know the relative magnitudes of these effects (the parameter values). In practice they would have to be estimated, which requires fitting a model to observed data.
Write a set of first-order linear equations describing the relationship between each pair of variables. (Mathematical reminder: simple rate of change is a first derivative, so equations describing it are first-order, like the example equation given above. If we wanted to describe the rate of change of the rate of change, we would use a second derivative, and that would involve a second-order differential equation. A linear equation in variable X uses only the first power of X and does not contain terms like 3X², which in PLAS would be written 3 X^2. Keep things simple for now!) You don't need to use names like X1, but can name your variables almost as you like (I used the dummy names given in the figure: Gene_1, etc.!). Start with simple parameter values.
Remember that the change in concentration of a reactant may have two terms: a promotion and an inhibition term, which are added together. For example, X1' = a X2 - b X1 indicates that X1 is being produced from reactant X2 at a rate that decreases as the product accumulates (assuming that a and b are positive constants).
Save your file as clock.plc.
Now choose Analysis/Run. View the resulting plot. Be sure you understand qualitatively the behavior of each of the variables. Could these variables ever fluctuate in a cyclical manner, using the system you have specified and assigning some special combination of parameters (rate constants)? If you can't answer this by thinking about it, check by experimenting with different parameter settings without changing the model. With each change, rerun the analysis. Note: sometimes you will want to adjust the vertical scale. To do this, double-click on the highest label on the Y axis at the left side of the plot, and enter a value that seems more appropriate.
Still trying to produce cyclical behavior, modify the model by adding constant terms to the rates. For example, you might change X2' = b X1 to X2' = a + b X1. Also see whether reversing the signs of the terms can help; for example we might pretend that Protein 1 inhibits the expression of Gene 2, contrary to the model shown. Comment on your results. Note that you can copy and save the plots generated by PLAS.
It is possible to produce a cyclical plot with the preceding methods and the right parameter values. However, you are free to introduce some more complexity into the model by pretending that the figure above doesn't show all of the relationships. Still sticking to first-order equations, modify the equations so that the left-hand side (for short, the LHS) term (in general, the first derivative of one of the components) is a function of more than one of the quantities, written on the RHS. For example we might write X2' = a X1 + b5 X3, or X2' = a X1 b X3 (the product instead of the sum). But don't try to be too fancy -- it will be nearly impossible to see why your equations generate the plots, let alone imagine any biological parallel to them. You'll learn more if, rather than just playing with numbers, you think of a biochemical scenario, attempt to predict the behavior of the system, model it, and see whether your predictions are borne out. Describe one such experiment.
You may also introduce terms of higher power, such as X^2 - a X^3. But keep in mind that our goal is to model a cyclic system, one that returns to the same state for every variable at a fixed period of time. Describe one of your experiments.
Extra credit possible Introduce higher-order derivatives; for example the LHS of one of your equations might be X1''. Recall from your high-school physics: if X1 represented distance, X1'' would represent acceleration. Again, don't just write complex systems of equations in order to see if you can produce interesting behavior by chance. Write one or two simple second-order systems with some hypothesis about the behavior you are aiming at, and describe the results of one.
At right is one model for the circadian clock cycle. The lambdas are constants that describe the relationship between the concentration and the rate of change of a reactant molecule, while the Rs represent the influence on that rate of change by the other reactants in the system. The Ms refer to the genes (Gene_1 and Gene_2) and the Ps to the proteins in the scheme above. The LHS terms are first derivatives, just as in the equations above: dM₁/dt is just M₁'. The h is a scaling constant, which you may assign the value 0.0001. The n is a cooperativity constant whose value must be at least 5 for the circadian-clock model to work.
Write a PLAS model for this system. To represent the lambda, n, and h terms use PLAS's system for defining constants, described in the Syntax Reference section of the Help. For example, write lambda_M1 = 0.5 and then in your differential equation for M₁, use lambda_M1 instead of the numerical value. This is just a convenient use of abstraction -- it allows us to change the value of a constant while leaving the equations unchanged.
With the first equation you will probably get an error message from PLAS: Rational expression in a GMA. All you need to do is change the solver (numerical calculation engine) that PLAS is using. Choose the Options menu. In the dialog box, click the Solver tab, and choose the Adams/BDF (LSODA) button, then click OK.
Try to find parameter values such that the system cycles at about 24 h. Make and save copies of phase plots P1 vs. P2 and M1 vs. P1. To change the variables used in a phase plot, double-click on the name of the variable in the plot. Explain why, in a cycling system, the phase plot takes this form (compare to a phase plot from a noncyclic system such as those you modeled in the earlier steps). Plot the time series, describe the system behavior that it illustrates, and explain concisely whether the behavior of each of the variables seems consistent with the relationships described in the model. For example, the first equation indicates that the rate of degradation of M1 is proportional to its concentration and its rate of production is inversely proportional to the concentration of P2. Is this reflected in the plot(s)?

Notes

Don't forget to supply the equations for the systems that you describe.
To turn your plots into images, choose Results/Copy, then open a graphics program such as Paint or IrfanView, paste into a window, and save your image as a .jpg or .gif image. Be sure to provide captions to images inserted into your report.