6.1: Relations
A relation in mathematics is a symbol that can be placed between two numbers (or variables) to create a logical statement (or open sentence). The main point here is that the insertion of a relation symbol between two numbers creates a statement whose value is either true or false. For example, we have previously seen the divisibility symbol (\(|\)) and noted the common error of mistaking it for the division symbol (\(/\)); one of these tells us to perform an arithmetic operation, the other asks us whether if such an operation were performed there would be a remainder. There are many other symbols that we have seen which have this characteristic, the most important is probably \(=\), but there are lots: \(\neq, <, \leq, >, ≥\) all work this way – if we place them between two numbers we get a Boolean thing, it’s either true or false. If, instead of numbers, we think of placing sets on either side of a relation symbol, then \(=\), \(⊆\) and \(⊇\) are valid relation symbols. If we think of placing logical expressions on either side of a relation then, honestly, any of the logical symbols is a relation, but we normally think of \(∧\) and \(∨\) as operators and give things like \(≡\), \(\implies\) and \(\iff\) the status of relations.
In the examples we’ve looked at the things on either side of a relation are of the same type. This is usually, but not always, the case. The prevalence of relations with the same kind of things being compared has even lead to the aphorism “Don’t compare apples and oranges.” Think about the symbol \(∈\) for a moment. As we’ve seen previously, it isn’t usually appropriate to put sets on either side of this, we might have numbers or other objects on the left and sets on the right. Let’s look at a small example. Let \(A = \{1, 2, 3, a, b\}\) and let \(B = \{\{1, 2, a\}, \{1, 3, 5, 7, . . .\}, \{1\}\}\). The “element of” relation, \(∈\), is a relation from \(A\) to \(B\).
A diagram such as we have given in Figure \(6.1.1\) seems like a very natural thing. Such pictures certainly give us an easy visual tool for thinking about relations. But we should point out certain hidden assumptions. First, they’ll only work if we are dealing with finite sets, or sets like the odd numbers in our example (sets that are infinite but could in principle be listed). Second, by drawing the two sets separately, it seems that we are assuming they are not only different, but disjoint. The sets not only need not be disjoint, but often (most of the time!) we have relations that go from a set to itself so the sets in a picture like this may be identical. In Figure \(6.1.2\) we illustrate the divisibility relation on the set of all divisors of \(6\) — this is an example in which the sets on either side of the relation are the same. Notice the linguistic distinction, we can talk about either “a relation from \(A\) to \(B\)” (when there are really two different sets) or “a relation on \(A\)” (when there is only one).
Purists will note that it is really inappropriate to represent the same set in two different places in a Venn diagram. The diagram in Figure \(6.1.2\) should really look like this:
Indeed, this representation is definitely preferable, although it may be more crowded. A picture such as this is known as the directed graph (a.k.a. digraph ) of the relation.
Recall that when we were discussing sets we said the best way to describe a set is simply to list all of its elements. Well, what is the best way to describe a relation? In the same spirit, it would seem we should explicitly list all the things that make the relation true. But it takes a pair of things, one to go on the left side and one to go on the right, to make a relation true (or for that matter false!). Also, it should be evident that order is important in this context, for example \(2 < 3\) is true but \(3 < 2\) isn’t. The identity of a relation is so intimately tied up with the set of ordered pairs that make it true, that when dealing with abstract relations we define them as sets of ordered pairs.
Given two sets, \(A\) and \(B\), the Cartesian product of \(A\) and \(B\) is the set of all ordered pairs \((a, b)\) where \(a\) is in \(A\) and \(b\) is in \(B\). We denote the Cartesian product using the symbol \(×\).
\(A × B = \{(a, b) a ∈ A ∧ b ∈ B\}\)
From here on out in your mathematical career, you’ll need to take note of the context that the symbol \(×\) appears in. If it appears between numbers go ahead and multiply, but if it appears between sets you’re doing something different – forming the Cartesian product.
The familiar \(x\)–\(y\) plane, is often called the Cartesian plane. This is done for two reasons. Rene Descartes, the famous mathematician and philosopher, was the first to consider coordinatizing the plane and thus is responsible for our current understanding of the relationship between geometry and algebra. Rene Descartes’ name is also memorialized in the definition of the Cartesian product of sets, and the plane is nothing more than the product \(\mathbb{R} × \mathbb{R}\). Indeed, the plane provided the very first example of the concept that was later generalized to the Cartesian product of sets.
Suppose \(A = \{1, 2, 3\}\) and \(B = \{a, b, c\}\). Is \((a, 1)\) in the Cartesian product \(A × B\)? List all elements of \(A × B\).
In the abstract, we can define a relation as any subset of an appropriate Cartesian product. So an abstract relation \(\text{R}\) from a set \(A\) to a set \(B\) is just some subset of A×B. Similarly, a relation \(\text{R}\) on a set \(S\) is defined by a subset of \(S × S\). This definition looks a little bit strange when we apply it to an actual (concrete) relation that we already know about. Consider the relation “less than.” To describe “less than” as a subset of a Cartesian product we must write
\(< = \{(x, y) ∈ \mathbb{R} × \mathbb{R} y − x ∈ \mathbb{R}^+\}\).
This looks funny.
Also, if we have defined some relation \(\text{R} ⊆ A × B\), then in order to say that a particular pair, \((a, b)\), of things make the relation true we have to write
\(a\text{R}b\).
This looks funny too.
Despite the strange appearances, these examples do express the correct way to deal with relations.
Let’s do a completely made-up example. Suppose \(A\) is the set \(\{a, e, i, o, u\}\) and \(B\) is the set \(\{r, s, t, l, n\}\) and we define a relation from \(A\) to \(B\) by
\(R = \{(a, s),(a, t),(a, n),(e, t),(e, l),(e, n),(i, s),(i, t),(o, r),(o, n),(u, s)\}\).
Then, for example, because \((e, t) ∈ \text{R}\) we can write \(e\text{R}t\). We indicate the negation of the concept that two elements are related by drawing a slash through the name of the relation, for example the notation \(\neq\) is certainly familiar to you, as is \(\nless\) (although in this latter case we would normally write \(≥\) instead). We can denote the fact that \((a, l)\) is not a pair that makes the relation true by writing \(a \not \text{R} l\).
We should mention another way of visualizing relations. When we are dealing with a relation on \(\mathbb{R}\), the relation is actually a subset of \(\mathbb{R} × \mathbb{R}\), that means we can view the relation as a subset of the \(x\)–\(y\) plane. In other words, we can graph it. The graph of the “\(<\)” relation is given in Figure \(6.1.3\).
A relation on any set that is a subset of \(\mathbb{R}\) can likewise be graphed. The graph of the “\(|\)” relation is given in Figure \(6.1.4\).
Eventually, we will get around to defining functions as relations that have a certain nice property. For the moment, we’ll just note that some of the operations that you are used to using with functions also apply with relations. When one function “undoes” what another function “does” we say the functions are inverses. For example, the function \(f(x) = 2x\) (i.e. doubling) and the function \(g(x) = \dfrac{x}{2}\) (halving) are inverse functions because, no matter what number we start with, if we double it and then halve that result, we end up with the original number. The inverse of a relation \(\text{R}\) is written \(\text{R}^{−1}\) and it consists of the reversals of the pairs in \(\text{R}\),
\(\text{R}^{−1} = \{(b, a) (a, b) ∈ \text{R}\}\).
This can also be expressed by writing
\(b\text{R}^{−1}a \iff a\text{R}b.\)
The process of “doing one function and then doing another” is known as functional composition. For instance, if \(f(x) = 2x + 1\) and \(g(x) = \sqrt{x}\), then we can compose them (in two different orders) to obtain either \(f(g(x)) = 2 \sqrt{x} + 1\) or \(g(f(x)) = \sqrt{2x + 1}\). When composing functions there is an “intermediate result” that you get by applying the first function to your input, and then you calculate the second function’s value at the intermediate result. (For example, in calculating \(g(f(4))\) we get the intermediate result \(f(4) = 9\) and then we go on to calculate \(g(9) = 3\).)
The definition of the composite of two relations focuses very much on this idea of the intermediate result. Suppose \(\text{R}\) is a relation from \(A\) to \(B\) and \(\text{S}\) is a relation from \(B\) to \(C\) then the composite \(\text{S} ◦ \text{R}\) is given by
\(\text{S} ◦ \text{R} = \{(a, c) ∃b ∈ B,(a, b) ∈ \text{R} ∧ (b, c) ∈ \text{S}\}\).
In this definition, \(b\) is the “intermediate result,” if there is no such b that serves to connect \(a\) to \(c\) then \((a, c)\) won’t be in the composite. Also, notice that this is the composition \(\text{R}\) first, then \(\text{S}\), but it is written as \(\text{S} ◦ \text{R}\) – watch out for this! The compositions of relations should be read from right to left. This convention makes sense when you consider functional composition, \(f(g(x))\) means \(g\) first, then \(f\) so if we use the “little circle” notation for the composition of relations we have \(f ◦g(x) = f(g(x))\) which is nice because the symbols \(f\) and \(g\) appear in the same order. But beware! there are atavists out there who write their compositions the other way around.
You should probably have a diagram like the following in mind while thinking about the composition of relations. Here, we have the set \(A = \{1, 2, 3, 4\}\), the set \(B\) is \(\{a, b, c, d\}\) and \(C = \{w, x, y, z\}\). The relation \(\text{R}\) goes from \(A\) to \(B\) and consists of the following set of pairs,
\(R = \{(1, a),(1, c),(2, d),(3, c),(3, d)\}\).
And
\(S = \{(a, y),(b, w),(b, x),(b, z)\}\).
Notice that the composition \(\text{R} ◦ \text{S}\) is impossible (or, more properly, it is empty). Why? What is the (only) pair in the composition \(\text{S} ◦ \text{R}\)?
Exercises:
The lexicographic order, \(<_{\text{lex}}\), is a relation on the set of all words, where \(x <_{\text{lex}} y\) means that \(x\) would come before \(y\) in the dictionary. Consider just the three letter words like “iff”, “fig”, “the”, et cetera. Come up with a usable definition for \(x_1x_2x_3 <_{\text{lex}} y_1y_2y_3\).
What is the graph of “\(=\)” in \(\mathbb{R} × \mathbb{R}\)?
The inverse of a relation \(\text{R}\) is denoted \(\text{R}^{−1}\). It contains exactly the same ordered pairs as \(\text{R}\) but with the order switched. (So technically, they aren’t exactly the same ordered pairs . . . )
\(\text{R}^{−1} = \{(b, a) (a, b) ∈ \text{R}\}\)
Define a relation \(\text{S}\) on \(\mathbb{R} × \mathbb{R}\) by \(\text{S} = \{(x, y) y = \sin x\}\). What is \(\text{S}^{−1}\)? Draw a single graph containing \(\text{S}\) and \(\text{S}^{−1}\).
 If we think of undoing the process of putting on our socks and shoes (that’s socks first, then shoes) we have to first remove our shoes, then take off our socks.
The socks and shoes rule is valid for relations as well.
Prove that \((\text{S} ◦ \text{R})^{−1} = \text{R}^{−1} ◦ \text{S}^{−1}\).


13.1: Cancer Overview
How Cancer Arises
Cancer is caused by changes to genes that control the way our cells function, especially how they grow and divide.
Genetic changes that cause cancer can be inherited from our parents. They can also arise during a person’s lifetime as a result of errors that occur as cells divide or because of damage to DNA caused by certain environmental exposures. Cancer-causing environmental exposures include substances, such as the chemicals in tobacco smoke, and radiation, such as ultraviolet rays from the sun.
When Cancer Spreads
A cancer that has spread from the place where it first started to another place in the body is called metastatic cancer. The process by which cancer cells spread to other parts of the body is called metastasis .
Metastatic cancer has the same name and the same type of cancer cells as the original, or primary, cancer.
Under a microscope, metastatic cancer cells generally look the same as cells of the original cancer. Moreover, metastatic cancer cells and cells of the original cancer usually have some molecular features in common, such as the presence of specific chromosome changes.
Tissue Changes That Are Not Cancer
Not every change in the body’s tissues is cancer. Some tissue changes may develop into cancer if they are not treated, however. Here are some examples of tissue changes that are not cancer but, in some cases, are monitored:
Hyperplasia occurs when cells within a tissue divide faster than normal and extra cells build up, or proliferate. However, the cells and the way the tissue is organized look normal under a microscope. Hyperplasia can be caused by several factors or conditions, including chronic irritation.
Dysplasia is a more serious condition than hyperplasia. In dysplasia, there is also a buildup of extra cells. But the cells look abnormal and there are changes in how the tissue is organized. In general, the more abnormal the cells and tissue look, the greater the chance that cancer will form.
Some types of dysplasia may need to be monitored or treated. An example of dysplasia is an abnormal mole (called a dysplastic nevus) that forms on the skin. A dysplastic nevus can turn into melanoma, although most do not.
- What is Cancer?. Authored by : National Cancer Institute. Provided by : National Institutes of Health. Located at : https://www.cancer.gov/about-cancer/understanding/what-is-cancer . License : Public Domain: No Known Copyright
- Cell image. Authored by : National Institutes of Health. Located at : https://commons.wikimedia.org/wiki/File:Lung_cancer_cell_during_cell_division-NIH.jpg . License : Public Domain: No Known Copyright


21.2: Eigenfunctions and Eigenvalues
As we have already seen, an eigenfunction of an operator \(\hat{A}\) is a function \(f\) such that the application of \(\hat{A}\) on \(f\) gives \(f\) again, times a constant:
\[ \hat{A} f = k f, \label{22.2.1} \]
where \(k\) is a constant called the eigenvalue.
When a system is in an eigenstate of observable \(A\) (i.e., when the wave function is an eigenfunction of the operator \(\hat{A}\)) then the expectation value of \(A\) is the eigenvalue of the wave function. Therefore:
\[ \hat{A} \psi({\bf r}) = a \psi({\bf r}), \label{22.2.2} \]
then:
\[\begin{equation} \begin{aligned}\langle A \rangle &= \int \psi^{*}({\bf r}) \hat{A} \psi({\bf r}) d{\bf r} \\ &= \int \psi^{*}({\bf r}) a \psi({\bf r}) d{\bf r} \\ &= a \int \psi^{*}({\bf r}) \psi({\bf r}) d{\bf r} = a, \end{aligned} \end{equation} \label{22.2.3} \]
which implies that:
\[ \int \psi^{*}({\bf r}) \psi({\bf r}) d{\bf r} = 1. \label{22.2.4} \]
This property of wave functions is called normalization, and in the one-electron TISEq guarantees that the maximum probability of finding an electron over the entire space is one. 1
A unique property of quantum mechanics is that a wave function can be expressed not just as a simple eigenfunction, but also as a combination of several of them. We have in part already encountered such property in the previous chapter, where complex hydrogen orbitals have been combined to form corresponding linear ones. As a general example, let us consider a wave function written as a linear combination of two eigenstates of \(\hat{A}\), with eigenvalues \(a\) and \(b\):
\[ \psi = c_a \psi_a + c_b \psi_b, \label{22.2.5} \]
where \(\hat{A} \psi_a = a \psi_a\) and \(\hat{A} \psi_b = b \psi_b\). Then, since \(\psi_a\) and \(\psi_b\) are orthogonal and normalized (usually abbreviated as orthonormal), the expectation value of \(A\) is:
\[\begin{equation} \begin{aligned}\langle A \rangle &= \int \psi^{*} \hat{A} \psi d{\bf r} \\ &= \int \left[ c_a \psi_a + c_b \psi_b \right]^{*} \hat{A} \left[ c_a \psi_a + c_b \psi_b \right] d{\bf r}\\ &= \int \left[ c_a \psi_a + c_b \psi_b \right]^{*} \left[ a c_a \psi_a + b c_b \psi_b \right] d{\bf r}\\ &= a \vert c_a\vert^2 \int \psi_a^{*} \psi_a d{\bf r} + b c_a^{*} c_b \int \psi_a^{*} \psi_b d{\bf r} + a c_b^{*} c_a \int \psi_b^{*} \psi_a d{\bf r} + b \vert c_b\vert^2 \int \psi_b^{*} \psi_b d{\bf r}\\ &= a \vert c_a\vert^2 + b \vert c_b\vert^2. \end{aligned} \end{equation} \label{22.2.6} \]
This result shows that the average value of \(A\) is a weighted average of eigenvalues, with the weights being the squares of the coefficients of the eigenvectors in the overall wavefunction. 2
- Imposing the normalization condition is the best way to find the constant \(A\) in the solution of the TISEq for the particle in a box, a topic that we delayed in chapter 20 .
- This section was adapted in part from Prof. C. David Sherrill’s A Brief Review of Elementary Quantum Chemistry Notes available here .


9.3: Whole Genome Methods and Industrial Applications
- Explain the uses of genome-wide comparative analyses
- Summarize the advantages of genetically engineered pharmaceutical products
Advances in molecular biology have led to the creation of entirely new fields of science. Among these are fields that study aspects of whole genomes, collectively referred to as whole-genome methods. In this section, we’ll provide a brief overview of the whole-genome fields of genomics, transcriptomics, and proteomics.
Genomics, Transcriptomics, and Proteomics
The study and comparison of entire genomes, including the complete set of genes and their nucleotide sequence and organization, is called genomics. This field has great potential for future medical advances through the study of the human genome as well as the genomes of infectious organisms. Analysis of microbial genomes has contributed to the development of new antibiotics, diagnostic tools, vaccines, medical treatments, and environmental cleanup techniques.
The field of transcriptomics is the science of the entire collection of mRNA molecules produced by cells. Scientists compare gene expression patterns between infected and uninfected host cells, gaining important information about the cellular responses to infectious disease. Additionally, transcriptomics can be used to monitor the gene expression of virulence factors in microorganisms, aiding scientists in better understanding pathogenic processes from this viewpoint.
When genomics and transcriptomics are applied to entire microbial communities, we use the terms metagenomics and metatranscriptomics, respectively. Metagenomics and metatranscriptomics allow researchers to study genes and gene expression from a collection of multiple species, many of which may not be easily cultured or cultured at all in the laboratory. A DNA microarray (discussed in the previous section) can be used in metagenomics studies.
Another up-and-coming clinical application of genomics and transcriptomics is pharmacogenomics, also called toxicogenomics, which involves evaluating the effectiveness and safety of drugs on the basis of information from an individual’s genomic sequence. Genomic responses to drugs can be studied using experimental animals (such as laboratory rats or mice) or live cells in the laboratory before embarking on studies with humans. Changes in gene expression in the presence of a drug can sometimes be an early indicator of the potential for toxic effects. Personal genome sequence information may someday be used to prescribe medications that will be most effective and least toxic on the basis of the individual patient’s genotype.
The study of proteomics is an extension of genomics that allows scientists to study the entire complement of proteins in an organism, called the proteome. Even though all cells of a multicellular organism have the same set of genes, cells in various tissues produce different sets of proteins. Thus, the genome is constant, but the proteome varies and is dynamic within an organism. Proteomics may be used to study which proteins are expressed under various conditions within a single cell type or to compare protein expression patterns between different organisms.
The most prominent disease being studied with proteomic approaches is cancer, but this area of study is also being applied to infectious diseases. Research is currently underway to examine the feasibility of using proteomic approaches to diagnose various types of hepatitis, tuberculosis, and HIV infection, which are rather difficult to diagnose using currently available techniques. 1
A recent and developing proteomic analysis relies on identifying proteins called biomarkers, whose expression is affected by the disease process. Biomarkers are currently being used to detect various forms of cancer as well as infections caused by pathogens such as Yersinia pestis and Vaccinia virus . 2
Other “-omic” sciences related to genomics and proteomics include metabolomics, glycomics, and lipidomics, which focus on the complete set of small-molecule metabolites, sugars, and lipids, respectively, found within a cell. Through these various global approaches, scientists continue to collect, compile, and analyze large amounts of genetic information. This emerging field of bioinformatics can be used, among many other applications, for clues to treating diseases and understanding the workings of cells.
Additionally, researchers can use reverse genetics, a technique related to classic mutational analysis, to determine the function of specific genes. Classic methods of studying gene function involved searching for the genes responsible for a given phenotype. Reverse genetics uses the opposite approach, starting with a specific DNA sequence and attempting to determine what phenotype it produces. Alternatively, scientists can attach known genes (called reporter genes) that encode easily observable characteristics to genes of interest, and the location of expression of such genes of interest can be easily monitored. This gives the researcher important information about what the gene product might be doing or where it is located in the organism. Common reporter genes include bacterial lacZ , which encodes beta-galactosidase and whose activity can be monitored by changes in colony color in the presence of X-gal as previously described, and the gene encoding the jellyfish protein green fluorescent protein (GFP) whose activity can be visualized in colonies under ultraviolet light exposure (Figure \(\PageIndex{1}\)).
- How is genomics different from traditional genetics?
- If you wanted to study how two different cells in the body respond to an infection, what –omics field would you apply?
- What are the biomarkers uncovered in proteomics used for?
Why can some humans harbor opportunistic pathogens like Haemophilus influenzae , Staphylococcus aureus , or Streptococcus pyogenes , in their upper respiratory tracts but remain asymptomatic carriers, while other individuals become seriously ill when infected? There is evidence suggesting that differences in susceptibility to infection between patients may be a result, at least in part, of genetic differences between human hosts. For example, genetic differences in human leukocyte antigens (HLAs) and red blood cell antigens among hosts have been implicated in different immune responses and resulting disease progression from infection with H. influenzae .
Because the genetic interplay between pathogen and host may contribute to disease outcomes, understanding differences in genetic makeup between individuals may be an important clinical tool. Ecological genomics is a relatively new field that seeks to understand how the genotypes of different organisms interact with each other in nature. The field answers questions about how gene expression of one organism affects gene expression of another. Medical applications of ecological genomics will focus on how pathogens interact with specific individuals, as opposed to humans in general. Such analyses would allow medical professionals to use knowledge of an individual’s genotype to apply more individualized plans for treatment and prevention of disease.
With the advent of next-generation sequencing, it is relatively easy to obtain the entire genomic sequences of pathogens; a bacterial genome can be sequenced in as little as a day. 3 The speed and cost of sequencing the human genome has also been greatly reduced and, already, individuals can submit samples to receive extensive reports on their personal genetic traits, including ancestry and carrier status for various genetic diseases. As sequencing technologies progress further, such services will continue to become less expensive, more extensive, and quicker.
However, as this day quickly approaches, there are many ethical concerns with which society must grapple. For example, should genome sequencing be a standard practice for everybody? Should it be required by law or by employers if it will lower health-care costs? If one refuses genome sequencing, does he or she forfeit his or her right to health insurance coverage? For what purposes should the data be used? Who should oversee proper use of these data? If genome sequencing reveals predisposition to a particular disease, do insurance companies have the right to increase rates? Will employers treat an employee differently? Knowing that environmental influences also affect disease development, how should the data on the presence of a particular disease-causing allele in an individual be used ethically? The Genetic Information Nondiscrimination Act of 2008 (GINA) currently prohibits discriminatory practices based on genetic information by both health insurance companies and employers. However, GINA does not cover life, disability, or long-term care insurance policies. Clearly, all members of society must continue to engage in conversations about these issues so that such genomic data can be used to improve health care while simultaneously protecting an individual’s rights.
When Kayla described her symptoms, her physician at first suspected bacterial meningitis, which is consistent with her headaches and stiff neck. However, she soon ruled this out as a possibility because meningitis typically progresses more quickly than what Kayla was experiencing. Many of her symptoms still paralleled those of amyotrophic lateral sclerosis (ALS) and systemic lupus erythematosus (SLE), and the physician also considered Lyme disease a possibility given how much time Kayla spends in the woods. Kayla did not recall any recent tick bites (the typical means by which Lyme disease is transmitted) and she did not have the typical bull’s-eye rash associated with Lyme disease (Figure \(\PageIndex{2}\)). However, 20–30% of patients with Lyme disease never develop this rash, so the physician did not want to rule it out.
Kayla’s doctor ordered an MRI of her brain, a complete blood count to test for anemia, blood tests assessing liver and kidney function, and additional tests to confirm or rule out SLE or Lyme disease. Her test results were inconsistent with both SLE and ALS, and the result of the test looking for Lyme disease antibodies was “equivocal,” meaning inconclusive. Having ruled out ALS and SLE, Kayla’s doctor decided to run additional tests for Lyme disease.
- Why would Kayla’s doctor still suspect Lyme disease even if the test results did not detect Lyme antibodies in the blood?
- What type of molecular test might be used for the detection of blood antibodies to Lyme disease
Recombinant DNA Technology and Pharmaceutical Production
Genetic engineering has provided a way to create new pharmaceutical products called recombinant DNA pharmaceuticals. Such products include antibiotic drugs, vaccines, and hormones used to treat various diseases. Table \(\PageIndex{1}\) lists examples of recombinant DNA products and their uses.
For example, the naturally occurring antibiotic synthesis pathways of various Streptomyces spp., long known for their antibiotic production capabilities, can be modified to improve yields or to create new antibiotics through the introduction of genes encoding additional enzymes. More than 200 new antibiotics have been generated through the targeted inactivation of genes and the novel combination of antibiotic synthesis genes in antibiotic-producing Streptomyces hosts. 4
Genetic engineering is also used to manufacture subunit vaccines, which are safer than other vaccines because they contain only a single antigenic molecule and lack any part of the genome of the pathogen. For example, a vaccine for hepatitis B is created by inserting a gene encoding a hepatitis B surface protein into a yeast; the yeast then produces this protein, which the human immune system recognizes as an antigen. The hepatitis B antigen is purified from yeast cultures and administered to patients as a vaccine. Even though the vaccine does not contain the hepatitis B virus, the presence of the antigenic protein stimulates the immune system to produce antibodies that will protect the patient against the virus in the event of exposure. 5 6
Genetic engineering has also been important in the production of other therapeutic proteins, such as insulin, interferons, and human growth hormone, to treat a variety of human medical conditions. For example, at one time, it was possible to treat diabetes only by giving patients pig insulin, which caused allergic reactions due to small differences between the proteins expressed in human and pig insulin. However, since 1978, recombinant DNA technology has been used to produce large-scale quantities of human insulin using E. coli in a relatively inexpensive process that yields a more consistently effective pharmaceutical product. Scientists have also genetically engineered E. coli capable of producing human growth hormone (HGH), which is used to treat growth disorders in children and certain other disorders in adults. The HGH gene was cloned from a cDNA library and inserted into E. coli cells by cloning it into a bacterial vector. Eventually, genetic engineering will be used to produce DNA vaccines and various gene therapies, as well as customized medicines for fighting cancer and other diseases.
| Recombinant DNA Product | Application |
|---|---|
| Atrial natriuretic peptide | Treatment of heart disease (e.g., congestive heart failure), kidney disease, high blood pressure |
| DNase | Treatment of viscous lung secretions in cystic fibrosis |
| Erythropoietin | Treatment of severe anemia with kidney damage |
| Factor VIII | Treatment of hemophilia |
| Hepatitis B vaccine | Prevention of hepatitis B infection |
| Human growth hormone | Treatment of growth hormone deficiency, Turner’s syndrome, burns |
| Human insulin | Treatment of diabetes |
| Interferons | Treatment of multiple sclerosis, various cancers (e.g., melanoma), viral infections (e.g., Hepatitis B and C) |
| Tetracenomycins | Used as antibiotics |
| Tissue plasminogen activator | Treatment of pulmonary embolism in ischemic stroke, myocardial infarction |
- What bacterium has been genetically engineered to produce human insulin for the treatment of diabetes?
- Explain how microorganisms can be engineered to produce vaccines.
RNA Interference Technology
In chapter 10, we described the function of mRNA, rRNA, and tRNA. In addition to these types of RNA, cells also produce several types of small noncoding RNA molecules that are involved in the regulation of gene expression. These include antisense RNA molecules, which are complementary to regions of specific mRNA molecules found in both prokaryotes and eukaryotic cells. Non-coding RNA molecules play a major role in RNA interference (RNAi), a natural regulatory mechanism by which mRNA molecules are prevented from guiding the synthesis of proteins. RNA interference of specific genes results from the base pairing of short, single-stranded antisense RNA molecules to regions within complementary mRNA molecules, preventing protein synthesis. Cells use RNA interference to protect themselves from viral invasion, which may introduce double-stranded RNA molecules as part of the viral replication process (Figure \(\PageIndex{3}\)).
Researchers are currently developing techniques to mimic the natural process of RNA interference as a way to treat viral infections in eukaryotic cells. RNA interference technology involves using small interfering RNAs (siRNAs) or microRNAs (miRNAs) (Figure \(\PageIndex{4}\)). siRNAs are completely complementary to the mRNA transcript of a specific gene of interest while miRNAs are mostly complementary. These double-stranded RNAs are bound to DICER, an endonuclease that cleaves the RNA into short molecules (approximately 20 nucleotides long). The RNAs are then bound to RNA-induced silencing complex (RISC), a ribonucleoprotein. The siRNA-RISC complex binds to mRNA and cleaves it. For miRNA, only one of the two strands binds to RISC. The miRNA-RISC complex then binds to mRNA, inhibiting translation. If the miRNA is completely complementary to the target gene, then the mRNA can be cleaved. Taken together, these mechanisms are known as gene silencing.
Key Concepts and Summary
- The science of genomics allows researchers to study organisms on a holistic level and has many applications of medical relevance.
- Transcriptomics and proteomics allow researchers to compare gene expression patterns between different cells and shows great promise in better understanding global responses to various conditions.
- The various –omics technologies complement each other and together provide a more complete picture of an organism’s or microbial community’s ( metagenomics ) state.
- The analysis required for large data sets produced through genomics, transcriptomics, and proteomics has led to the emergence of bioinformatics .
- Reporter genes encoding easily observable characteristics are commonly used to track gene expression patterns of genes of unknown function.
- The use of recombinant DNA technology has revolutionized the pharmaceutical industry, allowing for the rapid production of high-quality recombinant DNA pharmaceuticals used to treat a wide variety of human conditions.
- RNA interference technology has great promise as a method of treating viral infections by silencing the expression of specific genes
Footnotes
- E.O. List, D.E. Berryman, B. Bower, L. Sackmann-Sala, E. Gosney, J. Ding, S. Okada, and J.J. Kopchick. “The Use of Proteomics to Study Infectious Diseases.” Infectious Disorders-Drug Targets (Formerly Current Drug Targets-Infectious Disorders ) 8 no. 1 (2008): 31–45.
- Mohan Natesan, and Robert G. Ulrich. “Protein Microarrays and Biomarkers of Infectious Disease.” International Journal of Molecular Sciences 11 no. 12 (2010): 5165–5183.
- D.J. Edwards, K.E. Holt. “Beginner’s Guide to Comparative Bacterial Genome Analysis Using Next-Generation Sequence Data.” Microbial Informatics and Experimentation 3 no. 1 (2013):2.
- Jose-Luis Adrio and Arnold L. Demain. “Recombinant Organisms for Production of Industrial Products.” Bioengineered Bugs 1 no. 2 (2010): 116–131.
- U.S. Department of Health and Human Services. “Types of Vaccines.” 2013. www.vaccines.gov/more_info/types/#subunit. Accessed May 27, 2016.
- The Internet Drug List. Recombivax . 2015. http://www.rxlist.com/recombivax-drug.htm . Accessed May 27, 2016.
Contributors and Attributions
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Nina Parker, (Shenandoah University), Mark Schneegurt (Wichita State University), Anh-Hue Thi Tu (Georgia Southwestern State University), Philip Lister (Central New Mexico Community College), and Brian M. Forster (Saint Joseph’s University) with many contributing authors. Original content via Openstax (CC BY 4.0; Access for free at https://openstax.org/books/microbiology/pages/1-introduction )


3: Some Teaser Problems
To illustrate some of the ways that we can employ a deliberate problem solving process, apply general heuristics, and look toward development of solution strategies, I offer some teaser problems. We’ll revisit these problems throughout the text where they serve as fodder for more targeted discussions of solution strategies. These problems vary in complexity and are in most cases fairly open ended. As we’ve already discussed, these are features of real problems and though we may feel a bit of anxiety about that, we should also realize that this presents opportunities for creativity and ingenuity.
3.1 Waterfowl easements
To have the greatest benefit to migratory waterfowl, is it better secure easements containing many small wetlands or fewer, larger wetlands?
One of the greatest threats to waterfowl populations is loss of habitat. Organizations like Ducks Unlimited and The Nature Conservancy have advocated for preservation and restoration of wetlands in key migration corridors and breeding areas. Among the strategies that organizations like this have used is acquisition of conservation easements, wherein a private land owner agrees to limit development on land containing quality habitat in exchange for benefits like management assistance and tax deductions. In practice, these organizations cannot accept all easement donations but need to prioritize those that will have the greatest long-term benefits to waterfowl. Thus the question: if all else is equal, is it better to prioritize a parcel with many small wetlands, or a parcel of the same size with fewer, large wetlands (Figure 3.1)?
3.2 Herbicide purchase
How much herbicide is needed to kill and suppress regrowth of invasive woody shrubs in a wooded urban park?
A common management problem in urban woodlands is the control of invasive or unwanted understory vegetation. In the midwestern USA, bush honeysuckles (genus Lonicera ) and European buck- thorn (genus Rhamnus ) are particularly challenging to manage, and the most effective means of control is usually with herbicide. The simplest approach for many of these woody invasives is to cut all the stems and paint stumps with a general herbicide like glyphosate. So what must we know and determine to decide upon the quantity of glyphosate required?
3.3 Deer-automobile collisions
How likely are collisions between automobiles and whitetail deer in your county, and how might they be influenced by changes in deer population management?
If you live in an urban environment without open spaces and wildlife corridors, the answer to this may be easy: zero likelihood. But in many areas both periurban and rural, deer are a familiar sight. In these settings, deer-automobile collisions may unfortunately be all too common. We might infer that the frequency of collisions would be greater where both deer and drivers are more numerous, but can we quantify this?
3.4 Prairie dog plague
Is it possible to prevent the spread of Sylvatic plague through prairie dog colonies?
Yersinia pestis is a flea-borne bacterium that causes Sylvatic plague, an often fatal disease that primarily affects rodents. The black-tailed prairie dog ( Cynomys ludivicianus ), already threatened by land-use change and hunting/poisoning within it’s native range in the North American Great Plains, periodically experiences plague epidemics that can decimate colonies. In addition to harming prairie dogs directly, however, plague can be transmitted both directly and through fleas from prairie dogs to their predators, including the endangered black-footed ferret. In isolated populations of prairie dogs, monitoring and active management can potentially be used to reduce the spread of plague, but designing such a strategy requires that we first understand the dynamics of disease transmission within colonies.
3.5 Forest fire losses
How much fuel management is optimal to minimize risk of fire damage in a forest managed for timber production?
In many western forests managed for timber production, historical fire exclusion has led to a buildup of fuel wood, leading to heightened risk of destructive fire. When fires do occur in these settings, high fuel density can lead to destructive crown fires that greatly diminish the value of the timber and can damage or destroy surrounding infrastructure. To combat this, forest managers can undertake fuel reduction treatments such as thinning and prescribed burning, and emergency managers can aggressively fight fires when they do start. However, such suppression and pre-suppression efforts can be costly and can have diminishing returns in terms of cost and re- sources (Figure 3.2). Is there an optimal amount of fuel reduction and suppression that provides some protection from risk while containing costs?
3.6 Maximized effluent
How much of phosphorus (P) can be discharged from point sources into an urban stream without exceeding total maximum daily loads (TMDLs)? \(^{1}\)
The U.S. Clean Water Act of 1972 establishes criteria for identify- ing polluted water bodies and designing plans to reduce pollutant loads and improve water quality. One of the measures available to water resource managers is the establishment of Total Maximum Daily Loads (TMDLs), which typically limit the permissible concentration of a pollutant in an impaired water body. If the goal in this case is to limit the amount of P delivered to downstream water bodies, we need to determine how much P constitutes the maximum allowable concentration.
3.7 Brook trout recruitment
Given data from four consecutive electrofishing passes in an isolated stream reach, what is the stream’s age- 0 brook trout population?
One method of estimating fish populations in streams is to isolate a stream reach with barriers upstream and downstream of and performing an electrofishing transit of the reach, collecting and measuring each fish retrieved. When extra detail is needed or fish are elusive, multiple passes through the reach may be required, and retrieved fish are removed to a live well or adjacent reach. Through this cost method, the change in catch between consecutive passes can be used to estimate the actual population within the reach. To isolate the population of age-0 (young of the year) fish, age must be assessed from each fish either directly\(^{2}\) or indirectly from the length distributions in the catch in each pass.
\(^{2}\) A nonlethal age estimate can be obtained by collecting and observing fish scales under a microscope
The problems above deal with a variety of practical issues that could be addressed by a natural resource professional. None of them are expressed explicitly as math problems, but quantitative methods could be instrumental in solving each of them. Indeed, this is characteristic of many real science and management problems. Before they become quantitative problems, they must be interpreted and carefully re-framed (Figure 3.3). This is not only an important part of the process, but in many cases is the most challenging and pivotal step. In practice, this must be part of the process of understanding the problem and perhaps even planning a solution strategy. In most cases, not enough information is given up front and the expectations for the kind of solution desired are vague. In this sense, they are ill-structured problems, requiring some digestion and careful re-phrasing before they can be understood as solvable quantitative problems.
Gathering the information deemed necessary to arrive at the solution is, of course, an essential step in the process. When possible, we should strive to include this step as part of the coursework. However the design of experiments and logistics of making novel field measurements can and should be addressed in their own courses, and we cannot hope to do justice to those concerns here. As a consequence, where the teaser problems are addressed here and in the chapters that follow, we’ll either work with hypothetical data or engage real data from the literature or from government documents and web resources.
Importantly, these problems also vary in the degree to which we could ever hope to know that we have the “right” answer. For example, It may indeed be possible to know exactly the maximum P load delivered in Problem 3.6 or the herbicide volume needed in Problem 3.2, provided that we had perfect information on stream discharge and stem basal area, respectively. However, for most of the other problems, confidence that we have a good solution must come from confidence that we have made well-reasoned interpretations and assumptions and used technically-correct manipulations and analysis. Thus, part of the burden of solving such a problem is articulating how the problem is interpreted and how chosen solution strategy follows from that interpretation. Each of these problems provides hints or implicit cues to what
kinds of strategies are appropriate to assemble a solution. For example, there are clear spatial elements to Problem 3.1 and Problem 3.2, but there may also be subtler spatial components in others. By spatial, I mean that we need to incorporate information about how big or how long something is into our solution. Addressing these elements of problems requires some tools from geometry and trigonometry, as well as perhaps the language and conventions of geography or cartography. Therefore, this book contains several chapters exploring the aspects of spatial reasoning that arise frequently in natural resources.
Spatial reasoning is the use of spatial information or relationships, like lengths, areas, volumes, or directions, in the solution of problems.
Not surprisingly, several of these problems point to issues of how many , particularly Problem 3.3 and Problem 3.7. In these and other problems, we may wish to characterize individuals or populations in terms of a representative value or distribution of values, or we may need to compare values or proportions. These tasks engage our experience with arithmetic, descriptive statistics, and probability, but in many cases also require that we are careful with units and that we can work efficiently with the extremely large or small numbers sometimes encountered in the sciences. For these issues, we have several chapters devoted to numerical reasoning.
Numerical reasoning , as used in this book, is the manipulation, characterization, comparison, and interpretation of numerical values (such as data) in the service of problem solving.
Aspects of most of our problems can be expressed in terms of the relationship between multiple variables, or relationships between cause and effect. For example, in Problem 3.5 we may need a way to represent the relationship between how much effort is made in fire-suppression activities and their cost. Since we don’t necessarily have an idea ahead of time about how much effort is appropriate, an algebraic expression relating the variable C (cost of suppression efforts) to the amount of suppression effort itself (call it S ) could stand in and allow us to employ algebraic reasoning as a means to a solution.
Algebraic reasoning is the use of generalized variables and formal relationships between them, rather than numbers, as a means of constraining solutions.
Finally, several of these problems imply that we seek an assessment or prediction of future, hypothetical, or unobservable quantities or events. For example, Problem 3.4 seems to ask whether something is even possible, even though we don’t know what that something is. In these cases, finding a useful solution may require modeling. As indicated in Chapter 1, a model is simplified representation of real systems and is constructed for the purpose of exploring cause and effect or functional relationships between variables as system conditions are changed. To model the spread of plague among prairie dogs, then, we may need a way to characterize the transfer of fleas between an infected individual and an uninfected, but susceptible individual. This type of reasoning will often require algebraic constructs and simplifying assumptions, and may therefore require competency in algebraic reasoning. Even so, however, some insights can be gained from model construction even if the detailed relationships between
Modeling in this book refers to the construction of a simplified representation of real or hypothesized systems, often described with one or more equations relating variables to one another and to system properties, and used to explore complex or unobservable phenomena or relationships.
variables aren’t formally articulated with equations. The Parts and Chapters that follow are elaborations and demonstrations of problem-solving strategies employing each of these types of reasoning. Not all will be applicable to every problem, and there isn’t necessarily a sequence of interdependence. Therefore, I hope that rather than reading the text in order, that you will consult it as the need for ideas and insight arises in problems you are presented with. I don’t provide complete solutions to any of the teaser problems above, but do use parts of them to illustrate concepts and strategies as they arise.
Which of the teaser problems outlined above do you find most interesting or compelling and why?
Using the table on solving ill-structured problems in Chapter 2 as a guide, attempt to Understand a teaser problem of your choice from the list above. Write your response to each of the questions under the Understand prompt, or write “n/a” if not applicable.
Try to write one or part of one of the teaser problems as a more conventional math problem, with only equations and no words. What is challenging about doing this?
Write a problem of your own in a format similar to the teaser problems above: a short title, a boldface question, and a short paragraph elaborating on the context or significance of the question. Choose a natural resource topic that interests you.