Growth and Mutation Project
Introduction
Most large mutations create an organism which is inviable, and never make it to maturity. Occasionally however, a mutant may not only become a viable organism, but may even out compete the wild type organism in its environment. In this project we will explore a model of this competition between a mutant and its wild type. We will also use this project to become acquainted with the Berkeley Madonna software and with the procedures for studying difference equations.
We will look at a generic bacteria with very simple growth dynamics. We will include no density dependence and an infinite supply of food. What we will focus on is the role of mutation in the fraction of mutants in the total population and how the differences in intrinsic growth rates and mutation rates determine the success of the mutant.
In the procedures outlined below, we will work through the steps necessary to find an updating function for the fraction of mutants in the population. We will then use the updating function to answer a series of questions about the role of mutation in populations.
Experimental Observation
1. In many simple one-on-one competition models, one population frequently loses out to the other.
2. Mutant frequently coexist with the wild type, with neither one going extinct.
3. Understanding invasions of mutants or other species into an environment can help with pest management, recolonization, and management of endangered species.
Procedure - A Test Run
We will begin by selecting a test case to work with. We will work through the test case with numbers, and then go through with variable and parameters to get the updating function.
Suppose two strains of bacteria grow in a culture. Both the wild-type W and the mutant M grow exponentially, but at different rates. With no mutation or reversion, the updating functions would be
Wt+1 = rWt
Mt+1 = sMt.
Here the growth rate of the wild-type is determined by r, while that of the mutant is determined by s. The wild-type bacteria and the mutants both have offspring of both types, depending on mutation and reversion (mutating back to the wild-type) rates. We denote u as the fraction of offspring of wild-type bacteria which are mutants and v as the fraction of offspring of mutants that are wild-type.
To begin, assume the following values,
W0 =4.0x10^6,
M0 =2.0x10^5,
r = 1.5,
s =2.0,
u=0.2, and
v=0.1.
We now compute the number of each type after one cycle of reproduction, mutation and reversion, and the fraction of mutants.
Step 1. How many wild-type bacteria will there be after reproduction but before mutation? How many mutant bacteria will there be after reproduction but before reversion?
Step 2. How many of the wild-type offspring will mutate? How many of the mutant offspring will revert to wild-type?
Step 3. What is the total number of wild-type bacteria after reproduction and mutation and reversion? What is the total number of mutants?
Step 4. What is the total number of bacteria? Why is this different from the initial number of total bacteria?
Step 5. What is the fraction of mutant bacteria in the population before and after one cycle? Is the fraction increasing?
Finding the Updating Function
Here we will rework the exact same steps from the previous section, keeping the parameters and variables as letters rather than using the numbers. This will give us a generic updating function that we can use for a variety of different parameters and initial conditions.
Step 1. How many wild-type bacteria will there be after reproduction but before mutation? How many mutant bacteria will there be after reproduction but before reversion?
Step 2. How many of the wild-type offspring will mutate? How many of the mutant offspring will revert to wild-type?
Step 3. What is the total number of wild-type bacteria after reproduction and mutation and reversion? What is the total number of mutants?
Step 4. What is the total number of bacteria?
Step 5. What is the fraction of mutant bacteria in the population after one cycle?
Step 6. In Step 5. you found an function for the new fraction of mutants as a function of the old numbers of mutants and wild-types. Convert this function to an updating function of the fraction of mutants, rather than the numbers of each. Remember that the fraction of mutants plusthe fraction of wild-types = 1 .
Simulating Difference Equations with Berkeley Madonna
Note: this is a general introduction to Berkeley Madonna and the equations are NOT the ones to use for the lab. In Berkeley Madonna, open a new equation window. The screen appears with several lines already in the window:
METHOD RK4
STARTTIME = 0
STOPTIME=10
DT = 0.02
The first line sets the method for solving any differential (not difference) equations to be used. For difference equations, this line does not affect the solution. The next two lines are mostly self-explanatory. Recall that the stop time should reflect the proper units for the problem. This will not affect the shape of the curve nor will with last line. The last line is used to set the time between measurements. For example, if we were measuring E.coli, we might note that it takes about 25 minutes between cell divisions. This might lead us to use DT =25/60 where time is measured in hours.
There are three types of equations will be adding to the existing lines. The first is the initial value of any variable we wish to update. As an example, we use the mutant equation.
init b=b0
We will define b0 in a minute. The second line will be an equation describing exactly how to update the value of M from one step to the next.
NEXT b=growthrate*b
This notation suggests that the ”NEXT” value of ”b” is the value we get by putting all of the current values of our variables and constants into the equation that follows. The last step is to define the values for all of the constants and parameters we have used. Using parameters and constants rather than numbers allows us to change those values more easily later on.
b0=2*10^5
growthrate=1.02
Now click on the ”run” button on the window. An overflow complaint may appear along with a graph. The complaint arises from the population becoming larger than the computer can handle. That is ok for now, just click ”no” and look at the graph. When we simulate the equation for the proportion of mutants no such complaint should appear.
For your simulations, you will want to use a variable for the fraction of mutants (maybe f for fraction, or p for proportion) andf write the updating function leaving the parameters are letters. This will make it easier to change them later.
Questions and Ponderings
Many of the questions will require that you produce many different graphs to answer the question. Once you have determined the answer, not all of the graphs will be needed in the write-up. Please do not use more than necessary to illustrate your answer.
1. Explain the range of realistic values for each of the parameters for growth and mutation. Make sure you use only realistic values in answering the questions below.
2. For the values we used in our initial update, find the steady states of the system. Plot the solution as a function of time for a few different initial conditions to determine the stability of each of the steady states. Use several different values of r.
3. Suppose r = s and u= 0. Explore what happens to the equilibria on the computer and explain what this means biologically. Use several different values of r.
4. Suppose s = r and u= v. Explore what happens to the equilibria on the computer and explain what this means biologically. Use several different values of r and ?.
5. Suppose r>s, v =0,and u>0. The result is called mutation-selection balance. Using the computer to explore several different cases, explain why the name is fitting.
Follow- up
1. What do you feel was the purpose of this lab?
2. What did you learn in this lab?
3. What changes would you make to this lab?
4. Please rate the difficulty of this lab on a scale of 1 to 10.
The Write-up
The write-up MUST be typed. Graphs and figures should appear in the text where relevant. Answers to questions do not need to be written in numbered format and complete sentences are needed for full and informative explanations.
The write-up should include an introduction, a section on methods, and a conclusion section. The goal of the write-up is for someone not associated with the class (or me a few months from now), to read the write-up and understand what the problems were, what was done to discover the answers, and what the results were. This should be done in as readable a fashion as possible.
This project is based on material in Fred Adler’s text Modeling the Dynamics of Life.