9.8: Molecular Orbital Theory
Make sure you thoroughly understand the following essential ideas
- In what fundamental way does the molecular orbital model differ from the other models of chemical bonding that have been described in these lessons?
- Explain how bonding and antibonding orbitals arise from atomic orbitals, and how they differ physically.
- Describe the essential difference between a sigma and a pi molecular orbital.
- Define bond order , and state its significance.
- Construct a "molecular orbital diagram" of the kind shown in this lesson for a simple diatomic molecule, and indicate whether the molecule or its positive and negative ions should be stable.
The molecular orbital model is by far the most productive of the various models of chemical bonding, and serves as the basis for most quantiative calculations, including those that lead to many of the computer-generated images that you have seen elsewhere in these units. In its full development, molecular orbital theory involves a lot of complicated mathematics, but the fundamental ideas behind it are quite easily understood, and this is all we will try to accomplish in this lesson.
This is a big departure from the simple Lewis and VSEPR models that were based on the one-center orbitals of individual atoms. The more sophisticated hybridization model recognized that these orbitals will be modified by their interaction with other atoms. But all of these valence-bond models, as they are generally called, are very limited in their applicability and predictive power, because they fail to recognize that distribution of the pooled valence electrons is governed by the totality of positive centers.
Molecular Orbitals
Chemical bonding occurs when the net attractive forces between an electron and two nuclei exceeds the electrostatic repulsion between the two nuclei. For this to happen, the electron must be in a region of space which we call the binding region . Conversely, if the electron is off to one side, in an anti-binding region , it actually adds to the repulsion between the two nuclei and helps push them away.
The easiest way of visualizing a molecular orbital is to start by picturing two isolated atoms and the electron orbitals that each would have separately. These are just the orbitals of the separate atoms, by themselves, which we already understand. We will then try to predict the manner in which these atomic orbitals interact as we gradually move the two atoms closer together. Finally, we will reach some point where the internuclear distance corresponds to that of the molecule we are studying. The corresponding orbitals will then be the molecular orbitals of our new molecule.
To see how this works, we will consider the simplest possible molecule, \(\ce{H2^{+}}\). This is the hydrogen molecule ion, which consists of two nuclei of charge +1, and a single electron shared between them.
As two H nuclei move toward each other, the 1 s atomic orbitals of the isolated atoms gradually merge into a new molecular orbital in which the greatest electron density falls between the two nuclei. Since this is just the location in which electrons can exert the most attractive force on the two nuclei simultaneously, this arrangement constitutes a bonding molecular orbital . Regarding it as a three- dimensional region of space, we see that it is symmetrical about the line of centers between the nuclei; in accord with our usual nomenclature, we refer to this as a σ (sigma) orbital .
Bonding and Antibonding Molecular Orbitals
There is one minor difficulty: we started with two orbitals (the 1 s atomic orbitals), and ended up with only one orbital. Now according to the rules of quantum mechanics, orbitals cannot simply appear and disappear at our convenience. For one thing, this would raise the question of at just what internuclear distance do we suddenly change from having two orbitals, to having only one? It turns out that when orbitals interact, they are free to change their forms, but there must always be the same number. This is just another way of saying that there must always be the same number of possible allowed sets of electron quantum numbers.
How can we find the missing orbital? To answer this question, we must go back to the wave-like character of orbitals that we developed in our earlier treatment of the hydrogen atom. You are probably aware that wave phenomena such as sound waves, light waves, or even ocean waves can combine or interact with one another in two ways: they can either reinforce each other, resulting in a stronger wave, or they can interfere with and partially destroy each other. A roughly similar thing occurs when the “matter waves” corresponding to the two separate hydrogen 1 s orbitals interact; both in-phase and out-of-phase combinations are possible, and both occur. The in-phase, reinforcing interaction yields the bonding orbital that we just considered. The other, corresponding to out-of-phase combination of the two orbitals, gives rise to a molecular orbital that has its greatest electron probability in what is clearly the antibonding region of space. This second orbital is therefore called an antibonding orbital.
When the two 1 s wave functions combine out-of-phase, the regions of high electron probability do not merge. In fact, the orbitals act as if they actually repel each other. Notice particularly that there is a region of space exactly equidistant between the nuclei at which the probability of finding the electron is zero. This region is called a nodal surface , and is characteristic of antibonding orbitals. It should be clear that any electrons that find themselves in an antibonding orbital cannot possibly contribute to bond formation; in fact, they will actively oppose it.
We see, then, that whenever two orbitals, originally on separate atoms, begin to interact as we push the two nuclei toward each other, these two atomic orbitals will gradually merge into a pair of molecular orbitals, one of which will have bonding character, while the other will be antibonding. In a more advanced treatment, it would be fairly easy to show that this result follows quite naturally from the wave-like nature of the combining orbitals.
What is the difference between these two kinds of orbitals, as far as their potential energies are concerned? More precisely, which kind of orbital would enable an electron to be at a lower potential energy? Clearly, the potential energy decreases as the electron moves into a region that enables it to “see” the maximum amount of positive charge. In a simple diatomic molecule, this will be in the internuclear region— where the electron can be simultaneously close to two nuclei. The bonding orbital will therefore have the lower potential energy.
Molecular Orbital Diagrams
This scheme of bonding and antibonding orbitals is usually depicted by a molecular orbital diagram such as the one shown here for the dihydrogen ion H 2 + . Atomic valence electrons (shown in boxes on the left and right) fill the lower-energy molecular orbitals before the higher ones, just as is the case for atomic orbitals. Thus, the single electron in this simplest of all molecules goes into the bonding orbital, leaving the antibonding orbital empty.
Since any orbital can hold a maximum of two electrons, the bonding orbital in H 2 + is only half-full.  Although H 2 + is stable in this energetic sense, it happens to be an extremely reactive molecule— so much so that it even reacts with itself, so these ions are not commonly encountered in everyday chemistry.
Dihydrogen
If one electron in the bonding orbital is conducive to bond formation, might two electrons be even better? We can arrange this by combining two hydrogen atoms-- two nuclei, and two electrons. Both electrons will enter the bonding orbital, as depicted in the Figure.
We recall that one electron lowered the potential energy of the two nuclei by 270 kJ/mole, so we might expect two electrons to produce twice this much stabilization, or 540 kJ/mole.
Experimentally, one finds that it takes only 452 kJ to break apart a mole of hydrogen molecules. The reason the potential energy was not lowered by the full amount is that the presence of two electrons in the same orbital gives rise to a repulsion that acts against the stabilization. This is exactly the same effect we saw in comparing the ionization energies of the hydrogen and helium atoms.
Dihelium
With two electrons we are still ahead, so let’s try for three. The dihelium positive ion is a three-electron molecule. We can think of it as containing two helium nuclei and three electrons. This molecule is stable, but not as stable as dihydrogen; the energy required to break He 2 + is 301 kJ/mole. The reason for this should be obvious; two electrons were accommodated in the bonding orbital, but the third electron must go into the next higher slot— which turns out to be the sigma antibonding orbital. The presence of an electron in this orbital, as we have seen, gives rise to a repulsive component which acts against, and partially cancels out, the attractive effect of the filled bonding orbital.
Taking our building-up process one step further, we can look at the possibilities of combining to helium atoms to form dihelium. You should now be able to predict that He 2 cannot be a stable molecule; the reason, of course, is that we now have four electrons— two in the bonding orbital, and two in the antibonding orbital. The one orbital almost exactly cancels out the effect of the other. Experimentally, the bond energy of dihelium is only .084 kJ/mol; this is not enough to hold the two atoms together in the presence of random thermal motion at ordinary temperatures, so dihelium dissociates as quickly as it is formed, and is therefore not a distinct chemical species.
Diatomic molecules containing second-row atoms
The four simplest molecules we have examined so far involve molecular orbitals that derived from two 1 s atomic orbitals. If we wish to extend our model to larger atoms, we will have to contend with higher atomic orbitals as well. One greatly simplifying principle here is that only the valence-shell orbitals need to be considered. Inner atomic orbitals such as 1 s are deep within the atom and well-shielded from the electric field of a neighboring nucleus, so that these orbitals largely retain their atomic character when bonds are formed.
Dilithium
For example, when lithium, whose configuration is 1 s 2 2 s 1 , bonds with itself to form Li 2 , we can forget about the 1 s atomic orbitals and consider only the σ bonding and antibonding orbitals. Since there are not enough electrons to populate the antibonding orbital, the attractive forces win out and we have a stable molecule.
The bond energy of dilithium is 110 kJ/mole; notice that this value is less than half of the 270 kJ bond energy in dihydrogen, which also has two electrons in a bonding orbital. The reason, of course, is that the 2 s orbital of Li is much farther from its nucleus than is the 1 s orbital of H, and this is equally true for the corresponding molecular orbitals. It is a general rule, then, that the larger the parent atom, the less stable will be the corresponding diatomic molecule.
Lithium hydride
All the molecules we have considered thus far are homonuclear ; they are made up of one kind of atom. As an example of a heteronuclear molecule, let’s take a look at a very simple example— lithium hydride. Lithium hydride is a stable, though highly reactive molecule. The diagram shows how the molecular orbitals in lithium hydride can be related to the atomic orbitals of the parent atoms. One thing that makes this diagram look different from the ones we have seen previously is that the parent atomic orbitals have widely differing energies; the greater nuclear charge of lithium reduces the energy of its 1 s orbital to a value well below that of the 1 s hydrogen orbital.
There are two occupied atomic orbitals on the lithium atom, and only one on the hydrogen. With which of the lithium orbitals does the hydrogen 1 s orbital interact? The lithium 1 s orbital is the lowest-energy orbital on the diagram. Because this orbital is so small and retains its electrons so tightly, it does not contribute to bonding; we need consider only the 2 s orbital of lithium which combines with the 1 s orbital of hydrogen to form the usual pair of sigma bonding and antibonding orbitals. Of the four electrons in lithium and hydrogen, two are retained in the lithium 1 s orbital, and the two remaining ones reside in the σ orbital that constitutes the Li–H covalent bond.
The resulting molecule is 243 kJ/mole more stable than the parent atoms. As we might expect, the bond energy of the heteronuclear molecule is very close to the average of the energies of the corresponding homonuclear molecules. Actually, it turns out that the correct way to make this comparison is to take the geometric mean, rather than the arithmetic mean, of the two bond energies. The geometric mean is simply the square root of the product of the two energies.
The geometric mean of the H 2 and Li 2 bond energies is 213 kJ/mole, so it appears that the lithium hydride molecule is 30 kJ/mole more stable than it “is supposed” to be. This is attributed to the fact that the electrons in the 2σ bonding orbital are not equally shared between the two nuclei; the orbital is skewed slightly so that the electrons are attracted somewhat more to the hydrogen atom. This bond polarity , which we considered in some detail near the beginning of our study of covalent bonding, arises from the greater electron-attracting power of hydrogen— a consequence of the very small size of this atom. The electrons can be at a lower potential energy if they are slightly closer to the hydrogen end of the lithium hydride molecule. It is worth pointing out, however, that the electrons are, on the average, also closer to the lithium nucleus, compared to where they would be in the 2 s orbital of the isolated lithium atom. So it appears that everyone gains and no one loses here!
\(\Sigma\) and \(\pi\) orbitals
The molecules we have considered thus far are composed of atoms that have no more than four electrons each; our molecular orbitals have therefore been derived from s -type atomic orbitals only. If we wish to apply our model to molecules involving larger atoms, we must take a close look at the way in which p -type orbitals interact as well. Although two atomic p orbitals will be expected to split into bonding and antibonding orbitals just as before, it turns out that the extent of this splitting, and thus the relative energies of the resulting molecular orbitals, depend very much on the nature of the particular p orbital that is involved.
You will recall that there are three possible p orbitals for any value of the principal quantum number. You should also recall that p orbitals are not spherical like s orbitals, but are elongated, and thus possess definite directional properties. The three p orbitals correspond to the three directions of Cartesian space, and are frequently designated p x , p y , and p z , to indicate the axis along which the orbital is aligned. Of course, in the free atom, where no coordinate system is defined, all directions are equivalent, and so are the p orbitals. But when the atom is near another atom, the electric field due to that other atom acts as a point of reference that defines a set of directions. The line of centers between the two nuclei is conventionally taken as the x axis. If this direction is represented horizontally on a sheet of paper, then the y axis is in the vertical direction and the z axis would be normal to the page.
These directional differences lead to the formation of two different classes of molecular orbitals. The above figure shows how two p x atomic orbitals interact. In many ways the resulting molecular orbitals are similar to what we got when s atomic orbitals combined; the bonding orbital has a large electron density in the region between the two nuclei, and thus corresponds to the lower potential energy. In the out-of-phase combination, most of the electron density is away from the internuclear region, and as before, there is a surface exactly halfway between the nuclei that corresponds to zero electron density. This is clearly an antibonding orbital— again, in general shape, very much like the kind we saw in hydrogen and similar molecules. Like the ones derived from s -atomic orbitals, these molecular orbitals are σ ( sigma ) orbitals.
Sigma orbitals are cylindrically symmetric with respect to the line of centers of the nuclei; this means that if you could look down this line of centers, the electron density would be the same in all directions.
orbitals, we get the bonding and antibonding pairs that we would expect, but the resulting molecular orbitals have a different symmetry: rather than being rotationally symmetric about the line of centers, these orbitals extend in both perpendicular directions from this line of centers. Orbitals having this more complicated symmetry are called π ( pi ) orbitals. There are two of them, π y and π z differing only in orientation, but otherwise completely equivalent.
The different geometric properties of the π and σ orbitals causes the latter orbitals to split more than the π orbitals, so that the σ* antibonding orbital always has the highest energy. The σ bonding orbital can be either higher or lower than the π bonding orbitals, depending on the particular atom.
Second-Row Diatomics
If we combine the splitting schemes for the 2 s and 2 p orbitals, we can predict bond order in all of the diatomic molecules and ions composed of elements in the first complete row of the periodic table. Remember that only the valence orbitals of the atoms need be considered; as we saw in the cases of lithium hydride and dilithium, the inner orbitals remain tightly bound and retain their localized atomic character.
Dicarbon
Carbon has four outer-shell electrons, two 2 s and two 2 p . For two carbon atoms, we therefore have a total of eight electrons, which can be accommodated in the first four molecular orbitals. The lowest two are the 2 s -derived bonding and antibonding pair, so the “first” four electrons make no net contribution to bonding. The other four electrons go into the pair of pi bonding orbitals, and there are no more electrons for the antibonding orbitals— so we would expect the dicarbon molecule to be stable, and it is. (But being extremely reactive, it is known only in the gas phase.)
You will recall that one pair of electrons shared between two atoms constitutes a “single” chemical bond; this is Lewis’ original definition of the covalent bond. In C 2 there are two paris of electrons in the π bonding orbitals, so we have what amounts to a double bond here; in other words, the bond order in dicarbon is two.
Dioxygen
The electron configuration of oxygen is 1 s 2 2 s 2 2 p 4 . In O 2 , therefore, we need to accommodate twelve valence electrons (six from each oxygen atom) in molecular orbitals. As you can see from the diagram, this places two electrons in antibonding orbitals. Each of these electrons occupies a separate π* orbital because this leads to less electron-electron repulsion (Hund's Rule).
The bond energy of molecular oxygen is 498 kJ/mole. This is smaller than the 945 kJ bond energy of N 2 — not surprising, considering that oxygen has two electrons in an antibonding orbital, compared to nitrogen’s one.
The two unpaired electrons of the dioxygen molecule give this substance an unusual and distinctive property: O 2 is paramagneti c. The paramagnetism of oxygen can readily be demonstrated by pouring liquid O 2 between the poles of a strong permanent magnet; the liquid stream is trapped by the field and fills up the space between the poles.
Since molecular oxygen contains two electrons in an antibonding orbital, it might be possible to make the molecule more stable by removing one of these electrons, thus increasing the ratio of bonding to antibonding electrons in the molecule. Just as we would expect, and in accord with our model, O 2 + has a bond energy higher than that of neutral dioxygen; removing the one electron actually gives us a more stable molecule. This constitutes a very good test of our model of bonding and antibonding orbitals. In the same way, adding an electron to O 2 results in a weakening of the bond, as evidenced by the lower bond energy of O 2 – . The bond energy in this ion is not known, but the length of the bond is greater, and this is indicative of a lower bond energy. These two dioxygen ions, by the way, are highly reactive and can be observed only in the gas phase.


14.5B: Extracellular Immune Avoidance
- List the mechanisms that bacteria use for intracellular pathogenesis
Extracellular Immune Avoidance
A pathogen’s success depends on its ability to evade the host’s immune responses. Thus, pathogens have evolved several methods that allow them to successfully infect a host by evading the immune system’s detection and destruction. Bacteria usually overcome physical barriers by secreting enzymes that digest the barrier in the manner of a type II secretion system. They also use a type III secretion system that allows bacteria to insert a hallow tube, which provides proteins a direct route to enter the host cell. These proteins often shutdown the defenses of the host.
Some pathogens avoid the immune system by hiding within the cells of the host, a process referred to as intracellular pathogenesis. The pathogen hides inside the host cell where it is protected from direct contact with the complement, antibodies, and immune cells. A lot of pathogens release compounds that misdirect or diminish the host’s immune response. Some bacteria even form biofilms which protect them from the proteins and cells of the immune system. Many successful infections often involve biofilms. Some bacteria create surface proteins, such as Streptococcus, that will bind to antibodies making them ineffective.
Other pathogens invade the body by changing the non-essential epitopes on their surface rapidly while keeping the essential epitopes hidden. This is referred to as antigenic variation. HIV rapidly mutates so the proteins that are on its viral envelope, which are essential for its entry into the host’s target cell, are consistently changing. The constant change of these antigens is why vaccines have not been created. Another common strategy that is used is to mask antigens with host molecules in order to evade detection by the immune system. With HIV, the envelope covering the viron is created from the host cell’s outermost membrane making it difficult for the immune system to identify as a non-self structure.
Key Points
- Bacteria usually overcome physical barriers by secreting enzymes to digest the barrier in the manner of a type II secretion system.
- Some pathogens avoid the immune system by hiding within the cells of the host, a process referred to as intracellular pathogenesis.
- Other pathogens invade the body by changing the non-essential epitopes on their surface rapidly, while keeping the essential epitopes hidden.
Key Terms
- pathogen : Any organism or substance, especially a microorganism, capable of causing disease, such as bacteria, viruses, protozoa, or fungi. Microorganisms are not considered to be pathogenic until they have reached a population size that is large enough to cause disease.
- biofilm : A thin film of mucus created by and containing a colony of bacteria and other microorganisms.
- antigenic variation : The mechanism by which an infectious organism changes its surface proteins in favor of circumventing a host immune response.


8.14C: Hyperthermophiles from Submarine Volcanic Habitats
- Describe the metabolic processes used by hyperthermophiles found in submarine volcanic habitats
In biochemistry, chemosynthesis is the biological conversion of one or more carbon molecules (usually carbon dioxide or methane) and nutrients into organic matter using the oxidation of inorganic molecules (e.g. hydrogen gas, hydrogen sulfide) or methane as a source of energy. Chemoautotrophs, organisms that obtain carbon through chemosynthesis, are phylogenetically diverse. Groups that include conspicuous or biogeochemically-important taxa include the sulfur-oxidizing gamma and epsilon proteobacteria, the Aquificaeles, the methanogenic archaea and the neutrophilic iron-oxidizing bacteria.
A thermophile is an organism that thrives at relatively high temperatures, between 45 and 122 °C (113 and 252 °F). Thermophiles are found in various geothermally heated regions of the Earth, such as deep sea hydrothermal vents. As a prerequisite for their survival, thermophiles contain enzymes that can function at high temperatures. Some of these enzymes are used in molecular biology (for example, heat-stable DNA polymerases for PCR), and in washing agents.
Thermophiles are classified into obligate and facultative thermophiles: Obligate thermophiles (also called extreme thermophiles) require such high temperatures for growth, whereas facultative thermophiles (also called moderate thermophiles) can thrive at high temperatures, but also at lower temperatures (below 50°C). For example, Venenivibrio stagnispumantis gains energy by oxidizing hydrogen gas.
Many microorganisms in dark regions of the oceans also use chemosynthesis to produce biomass from single carbon molecules. Two categories can be distinguished. In the rare sites at which hydrogen molecules (H2) are available, the energy available from the reaction between CO2 and H2 (leading to production of methane, CH4) can be large enough to drive the production of biomass. Alternatively, in most oceanic environments, energy for chemosynthesis derives from reactions in which substances such as hydrogen sulfide or ammonia are oxidized to produce formaldehyde (which will be used to make carbohydrates) and solid globules of sulfur. This may occur with or without the presence of oxygen. In bacteria that can do this, such as purple sulfur bacteria, yellow globules of sulfur are present and visible in the cytoplasm.
Many chemosynthetic microorganisms are consumed by other organisms in the ocean, and symbiotic associations between chemosynthesizers and respiring heterotrophs are quite common. Large populations of animals can be supported by chemosynthetic secondary production at hydrothermal vents, methane clathrates, cold seeps, whale falls, and isolated cave water. Indeed, it has been hypothesized that chemosynthesis may support life below the surface of Mars, Jupiter’s moon Europa, and other planets. Giant tube worms use bacteria in their trophosome to react hydrogen sulfide with oxygen as a source of energy.
Key Points
- Chemosynthesis is the biological conversion of one or more carbon molecules and nutrients into organic matter using the oxidation of inorganic molecules (e.g. hydrogen gas, hydrogen sulfide ) or methane as a source of energy.
- The energy for chemosynthesis can be derived from hydrogen, hydrogen sulfide or ammonia.
- Many chemosynthetic microorganisms are consumed by other organisms in the ocean, and symbiotic associations between chemosynthesizers and respiring heterotrophs are quite common.
Key Terms
- hydrothermal vent : a hot spring, on the floor of the ocean, mostly along the central axes of the mid-ocean ridges, where heated fluids emerge from fissures in the Earth’s crust
- symbiotic : Of a relationship with mutual benefit between two individuals or organisms.
- chemosynthesis : The production of carbohydrates and other compounds from simple compounds such as carbon dioxide, using the oxidation of chemical nutrients as a source of energy rather than sunlight; it is limited to certain bacteria and fungi.


4.3: Protein-lipid Interactions
Lipids and proteins are essential elements of a living organism. Proteins, as one of the most abundant macromolecules of life, have countless functions ranging from catalyzing vital biological reactions to transporting nutrients. Likewise, lipids also play integral roles such as storing energy or being a major component of a membrane [12]. Despite their individual importance, interactions between these two molecules can provide functions that would not be possible individually. The greatest number of these interactions are seen in cell membranes, which are composed of a wide variety of lipids and proteins. While biological membranes elegantly fulfill many structural roles, the complexity of lipid-protein interactions facilitates a potentially stunning diversity of function and interaction.
Background
The crowded biological membrane (Figure \(\PageIndex{1}\)) provides countless interfaces for these interactions. In the membrane, the protein to lipid ratio is believed to be as high as 60:40 [15], and membrane proteins are believed to constitute an approximate 30% of the human genome [5]. These imply not only the individual importance of proteins and lipids to the function of biological membranes, but the potential complexity underlying interactions of proteins and lipids with each other. An understanding of the way proteins and lipids interact thus requires consideration of a variety of factors, from the bulk properties of the membrane to the intermolecular forces contributing to the behavior of individual lipids within the membrane.
Proteins found in a biological membrane have important roles in cell signaling, transduction and transportation [9]. One common way of classifying protein-lipid interactions is shown below:
Lipids in a biological membrane can be divided into three general groups [9].
- Bulk lipids: The lipids that form the main structure of the membrane without contacting the membrane proteins are called bulk lipids.
- Annular lipids: Some lipids surround membrane proteins and interact relatively non-specifically with them. These lipids are called the annular lipids, due to being similar to an annular shell (a ring-shaped layer in close contact with the protein surface).
- Non-annular lipids: These lipids interact specifically with some membrane proteins and are buried inside the protein structure. Non-annular lipids are often regarded similarly to protein cofactors--small molecules bound by a protein in its functional conformation.
Some of the vocabulary used to discuss membrane and protein properties is summarized in Figure \(\PageIndex{2}\):
- Integral membrane proteins : Proteins residing in at least one leaflet (layer) of the cell membrane.
- Transmembrane proteins: A class of integral membrane proteins spanning the full width of the cell membrane.
- Peripheral membrane proteins: Proteins that are transiently associated with the outer portion of one leaflet or an integral membrane protein.
- Hydrophobic thickness : The thickness of the portion of the membrane composed of the fatty acid lipid tails.
Bulk Lipid-Protein Interactions
Effect of Viscosity of the Membrane
The movement of any molecule or part of a molecule inside the membrane is influenced by the fluidity of this environment [9]. Every molecule experiences a frictional drag force during their movement inside the lipid bilayer structure of a biological membrane, and resistance of the fluid to this motion can be expressed in terms of the viscosity. By Stokes’ Law (Equation \ref{stokes}), the rate of particle motion can be expressed as a function of the drag force, membrane viscosity, and particle size [3]. This equation assumes the absence of turbulent flow (a good assumption for the comparatively calm cell membrane).
\[v = \dfrac{F}{6π \mu R} \label{stokes}\]
where
- \(F\): drag force (e.g. Newtons)
- \(\mu\): membrane viscosity (e.g. centipoise, Pascal*seconds)
- \(R\): particle size (e.g. meters, nanometers)
- \(v\): speed of particle motion (e.g. meters per second)
Larger particles, more viscous membranes, and more rapid motion thus all result in increased drag force. More viscous membranes can arise from the phase (e.g. liquid disordered, liquid ordered, or gel) of the lipids in the membrane; the phase is determined by the number of double bonds in the fatty acid tails of the lipids and the environmental temperature, with fewer double bonds or a temperature below the critical temperature for the lipids resulting in a more viscous membrane.
Movement of a protein in the membrane is dictated by the frictional resistance and the molecular restoring forces acting on it. A fluorescence polarization anisotropy study on the rotation of tryptophan (Trp) residues showed that the motion of Trp was affected by the viscosity of the lipid environment for small scale motion, and the amplitude of this motion increased with the temperature as the viscosity decreased. For example, at a viscosity of 1 centipoise (approximately the viscosity of water at room temperature and on the order of magnitude of the viscosity of the membrane) the authors expect local rotational motion of a given peptide within the protein to be largely determined by the surrounding peptide residues [17].
While the viscosity of the membrane can affect the rate of motion of the protein, the most thermodynamically favorable conformation of the protein should not depend on membrane viscosity [9]. This is because the thermodynamic properties of a system are inherently time-independent, but viscosity is a time-related parameter (since viscosity has units of pressure*time). This means a change in viscosity cannot directly increase or decrease the activation energy required to induce a conformational change--the most stable conformation of a protein is independent of the membrane viscosity. Changes in membrane viscosity can, however, change the amount of time required for the protein to assume this conformation. Additionally, if a change in membrane viscosity is the result of a change in temperature, the rate of protein function will be affected because rate constants are temperature-dependent [9]. This indirect effect can be worthwhile to consider in the study of isolated membrane proteins, which may take place at room temperature rather than at the temperature of the protein's natural environment.
Another important effect of membrane viscosity has also been revealed by molecular dynamics simulations. It has been illustrated that the effect of solvent viscosity on protein motion is important if the protein both directly contacts the solvent and has a rate of motion comparable to the environment. In other words, high frequency motion of a protein that does not overlap the solvent motion does not depend on the solvent’s viscosity [2]. Lipids are important in this context because viscosity of a biological membrane is mainly influenced by the types of lipids present, as the lipid composition is one of the main determinants of the fluidity.
Highly viscous membranes are indicative of increased ordering of the lipid fatty acid tails, often facilitated by interactions with smaller molecules such as sphingomyelin or cholesterol. This can result in increased membrane durability and impermeability. Additionally, the fluidity of the membrane environment surrounding a membrane protein can impact its function. For example, the human epidermal growth factor receptor (EGFR) was found to be non-functional in an environment in which both highly fluid liquid disordered and more viscous liquid ordered domains could exist. However, in a purely liquid disordered environment, EGFR retained its function, revealing a functional dependence on the fluidity of the membrane [3].
(See the main phase transitions page for more information on membrane fluidity.)
Localized areas of high membrane viscosity can also be indicative of the existence of lipid rafts, which are believed to play a role in protein isolation and function. Lipid rafts are transient sphingomyelin- and cholesterol-rich : (10-200 nm in diameter) within the cell membrane. The local abundance of sphingomyelin and/or cholesterol results in liquid-liquid phase separation, meaning lipid rafts are typically composed of the liquid ordered (L o ) phase. This means the lipids within rafts are more tightly packed and highly ordered than the bulk membrane, leading to increased local rigidity and decreased fluidity [18]. Some proteins can bind glycosphingolipids or sphingomyelin, which are involved in the recruitment of proteins to lipid rafts.
Effect of Membrane Curvature
Dynamic curvature of plasma or intercellular membrane can be dictated by the interactions between proteins and lipids. In fact, cells can employ various mechanisms to sense curvature as a way to create regions of active membrane trafficking. Lipid composition is a major influence on membrane curvature based on their chemical properties and/or the size of their headgroup. Presence of certain lipids are key to interact with certain peripheral membrane proteins in order to induce a necessary curvature. For example, phosphoinositides are required for budding of clathrin-coated vesicles, as the required machinery (e.g. coat proteins) can specifically bind to these lipids. The reason for this is that the headgroups of phosphoinositides are negatively charged, resulting in electrostatic repulsion that can contribute to membrane curvature [14].
Another study illustrated that NSA4(1-48), a key transmembrane protein that mediates replication of Dengue virus (a mosquito born single positive-stranded RNA virus), has an affinity for the convex face of highly curved regions of synthetic bilayer vesicles, as monitored by circular dichroism spectroscopy [8].
Similarly, a G protein-coupled serotonin receptor (5-HT 1A ) was found to function more rapidly in a more highly curved membrane [7]. This curvature effect could arise from a better match in the thickness of the hydrophobic portion of the membrane surrounding the protein, stabilizing the active protein form.
In addition to responding to changes in local curvature, membrane proteins can also influence the local curvature of the membrane. Some membrane proteins accomplish this by trafficking individual lipids across the membrane, resulting in uneven lipid compositions across the two membrane leaflets. The membrane must curve so that the leaflet with fewer lipids is on the interior of the curve in order to maintain optimal packing of the head groups [6]. Similarly, membrane leaflets containing lipids with different head group areas may also have some intrinsic curvature due to their differing surface areas [6]. In both cases, the integrity of the membrane as a barrier is facilitated by curvature, protecting the hydrophobic fatty acid tails from the aqueous environment and maintaining separation between the membrane bound compartment and its exterior.
Yet another example of the interplay between membrane proteins and membrane curvature is the process of vesicle formation. The ability to form vesicles is important in the trafficking of newly-formed proteins, the recycling of cell membrane components, and the intake of particles via endocytosis, among countless other cell functions. Vesicle formation involves complex regions of curvature, including low positive curvature when the vesicle is first beginning to form, greater positive curvature as the vesicle swells, and very high negative curvature where the vesicle meets the parent membrane. This complex curvature is facilitated in part by the membrane protein dynamin, which is responsible for creating the “neck” separating the parent membrane from the nearly fully-formed vesicle. This is a result of dynamin binding the membrane and taking on a helical conformation, physically forcing the parent membrane to assume a tube-like shape [14]. Coatomers are small proteins that may also play a role in vesicle formation by binding existing membrane proteins in a relatively confined area. The coatomers result in steric repulsion, leading to membrane curvature and budding [6].
The binding of amphipathic helices within the membrane is another common mechanism of protein-induced membrane curvature. As shown in the Figure below, one portion of the helix is hydrophobic, while the other is hydrophilic. The hydrophobic portion associates with the fatty acyl chains of the membrane, while the hydrophilic portion interacts with the head groups. This changes the area per head group for the bound leaflet of the membrane, and the membrane must curve in order to compensate.
Some of the other well understood ways that lipid-protein interaction relates to curvature of the membrane are discussed in membrane curvature in detail .
Effect of Changes in Membrane Thickness
An important feature of lipids that can influence protein function is the thickness of the bilayer. As shown in Figure \(\PageIndex{2}\), the hydrophobic thickness of a membrane is the distance between the hydrophilic headgroups of either leaflet, which is related to the length of the fatty acid tails of the lipids composing the membrane. Membrane proteins need to match the hydrophobic thickness of the acyl chains for two reasons. First, acyl chains and hydrophobic groups of the membrane proteins do not form hydrogen bonds with water, and tend to minimize their contact with it as this is more thermodynamically favorable (see hydrophobic effect ). When the hydrophobic thickness of the membrane and the membrane protein do not match, hydrophobic mismatch occurs. In order to minimize the exposure of hydrophobic portions of the lipid bilayer and transmembrane proteins to the aqueous environment, the bilayer can distort in various ways. It can stretch (Figure \(\PageIndex{5}\)), compress, or even tilt. Stretching involves an extension of the fatty acid chains of the lipids immediately surrounding the protein, providing a greater hydrophobic thickness to better match the size of the protein's hydrophobic domain. This minimizes the extent of hydrophobic membrane protein exposed to the aqueous environment, which is more energetically favorable. Conversely, compression involves a contraction of the fatty acid tails surrounding the protein.
Furthermore, the membrane proteins can aggregate (Figure \(\PageIndex{3}\)) to minimize the area exposed to water [16]. In addition to being more thermodynamically favorable (i.e. maximizing entropy by reducing the number of water molecules in highly ordered contact around hydrophobic residues), avoiding exposure of a transmembrane protein to aqueous environment is also important for optimal function because the hydrophobic side chains of the protein can change conformation when they contact water, distorting their native structure, and causing functional loss. The studies on these membrane distortions carried out with model membranes (artificial bilayers), show that the thickness of the acyl chain region is an essential part of protein function. For instance, sarcoplasmic reticulum Ca-ATPase, which is a protein important in calcium regulation in muscles, was shown to be greatly affected by the number of the carbons in the acyl chain, which determines the hydrophobic thickness of the membrane [16]. In addition, molecular dynamics simulations showed the possibility of protein aggregation (Figure \(\PageIndex{3}\)) due to hydrophobic mismatch [17].
Membrane proteins may also change conformation in response to changes in membrane tension, which can be indicative of changing external conditions necessitating a response from the cell. For example, osmotic swelling or shrinking of a cell will result in higher or lower membrane tension, respectively. Under increased tension, the bilayer will tend to thin, resulting in increased hydrophobic mismatch. One possibility to mitigate this is for the membrane conformation to change so that its hydrophobic domain thickness matches that of the transmembrane protein. Another possibility is that a conformational change in the membrane protein structure could relieve hydrophobic mismatch, making a response to membrane tension an effective gating mechanism. This is typically more energetically favorable in the case of significant membrane deformations, for which the energy required to change the membrane conformation to match the protein’s hydrophobic thickness would be greater than the cost of changing the protein’s conformation [11]. Such conformational changes include changes in the angle at which the transmembrane domain is tilted or unwinding of alpha-helical transmembrane domains [11]. The response of membrane proteins to hydrophobic mismatch arising from membrane tension is evident in the case of many bacterial mechanosensitive channels, which under higher membrane tension are more likely to be open [15].
For example, the lipid-protein interaction energy for a membrane protein with six helical transmembrane domains has been estimated to increase by approximately 10 -20 J/lipid molecule for a 10 Å hydrophobic mismatch, providing an appreciable driving force for either the membrane or protein to change its conformation [11]. At room temperature, the energetic cost of bending a lipid membrane is typically on the order of 10 -20 J, although this can change significantly based on the local temperature and lipid composition [24]. Hydrolyzing one ATP, a typical means of driving protein conformational changes, yields approximately 10 -19 J [6]. All these values are within similar orders of magnitude, suggesting the most effective method of resolving hydrophobic mismatch for a given membrane and protein may depend on both intrinsic and environmental factors.
Annular and Non-Annular Lipid-Protein Interactions
Protein-Lipid Interactions in Molecular Terms
In addition to the interactions of lipids and proteins mediated by the physical changes in the lipid environment, it is good to consider lipid-protein interactions on a molecular level. Types of lipids that were defined previously interact differently with proteins. Figure \(\PageIndex{4}\) is a basic representation of the interaction of bulk lipids and annular lipids with a protein surface. Lipids present in the first shell, having a higher level of interaction with the protein and more restricted motion, are the "annular lipids.” The second shell shows the bulk lipids.
Annular lipids and proteins weakly interact through van der Waals interactions, hydrogen bonding, and/or electrostatic interactions. These weak interactions allow annular lipids to be frequently exchanged with the lipids in the bulk bilayer due to the random motion expected in any fluid system, which implies that their interactions with proteins are fairly nonspecific. The level of interactions between annular lipids, which are highly dynamic, is an ongoing area of research as it still is a challenging task to analyze their structure. One example of interactions of such lipids with membrane proteins suggests that charges of annular lipids are important to the functionality of certain proteins. Accordingly, a type of ABC transporter protein, which can be involved in diseases such as multidrug resistance, preferred interacting with phosphatidylglycerol, a negatively charged phospholipid, and not with phosphatidylethanolamine, a zwitterionic phospholipid. This suggested the presence of higher affinity spots on the protein [1].
The neuronal protein syntaxin-1A, involved in the release of neurotransmitters, has also been shown to interact preferentially with negatively charged lipids [23]. Syntaxin-1A is found primarily in segregated clusters due to the local excess of a particular anionic lipid. The electrostatic interactions between the protein and lipid facilitate the formation of a protein microdomain without a change in the local phase of the surrounding lipids [23].
The existence of an annular shell of lipids around the transmembrane domain of a membrane protein arises from the distortion of the lipid fatty acid tails at the lipid-protein interface. The introduction of a solid, geometrically complex surface—the protein—into the membrane forces a rearrangement of the surrounding fatty acid tails in order to maintain the impermeability of the membrane. Without the lipid annulus, the tails would be able to pack only poorly around the protein, resulting in membrane leakiness. The entropic penalty of maintaining the ordered annular lipid shell is thus worthwhile in order to preserve the structural integrity of the membrane [15].
Unlike bulk and annular lipids, non-annular lipids interact specifically with proteins. These lipids are buried inside the protein, and are thought to be vital for optimal function. An example of such an interaction is found to be useful in bovine heart cytochrome c oxidase, on which 13 different types of lipids were suggested to have specific binding sites. It was also suggested that two palmitate tails of phosphatidylglycerols have a role in blocking the O 2 transfer pathway of this protein [19]. Lipid binding sites on the exterior of a similar protein are shown in the Figure below.
The interaction of non-annular lipids with proteins may also have implications in the detergent solubilization of membrane proteins—a common method used to isolate proteins from their native membranes to better understand the protein’s structure or function. If the detergent used is sufficiently similar to the annular lipids typically surrounding the protein, there is a possibility that the detergent could penetrate this lipid shell and interact with the protein during the solubilization process [13]. This could change the conformation of the protein, potentially impacting its function. For example, the amino acid transporter LeuT has been shown to bind detergent molecules in one of its active sites (which displace the lipids typically residing there). This reduces the activity of LeuT. However, when a larger detergent molecule was used, it was unable to penetrate the annular lipid shell during the solubilization process, resulting in a retention of the protein’s function [13]. This indicates that an understanding of the annular lipids in a protein's native membrane can be crucial to preserving both its structure and its function if solubilized.
Experimental Methods Used to Study Lipid-Protein Interactions
Some of the techniques used to study physical properties of lipid environment interacting with the membrane proteins include fluorescence spectroscopy, and electron paramagnetic resonance ( EPR ). There are methods utilized to understand interactions in molecular terms based on resolving the three-dimensional structures of proteins, which also helps clarify functions of some of the non-annular lipids [20]. X-Ray crystallography , NMR or EPR are among common tools analyzing lipid-protein interactions in molecular terms. Molecular dynamics simulations are another area of focus that can help researchers understand interactions of proteins and lipids both in molecular and physical terms.
The isolation of membrane proteins in proteoliposomes has also become a relatively common method of studying membrane protein function, including lipid-protein interactions. A proteoliposome is a small spherical vesicle into which a protein has been inserted. Proteoliposomes can be formed with known lipid compositions, allowing study of the effect of lipid composition--impacting membrane viscosity, phase separation, and membrane thickness-- on a protein’s function. The membrane of the proteoliposome can thus act as a host to a variety of membrane proteins. While this method cannot necessarily replicate the inherent crowding and complexity of a native cell membrane, it can provide an understanding of the relative membrane properties required for a protein’s function or the ligand affinity of the protein, among other things.
References
- Bechara, C., Noll, A., Morgner, N., Degiacomi, M., Tampe, R., & Robinson, C. (2015). A Subset of Annular Lipids is linked to the Flippase Activity of an ABC Transporter. Biophysical Journal.
- Brooks, C.L.; Karplus, M. Solvent effects on protein motion and protein effects on solvent motion. J. Mol. Biol. 208 (1989) 159–181.
- Coskun, Ü.; Grzybek, M.; Drechsel, D.; Simons, K. Regulation of Human EGF Receptor by Lipids. Proc. Natl. Acad. Sci. 2011 , 108 (22), 9044–9048.
- Deen, W. Analysis of Transport Phenomena, 2nd ed.; Oxford University Press, 2011.
- Fagerberg, L.; Jonasson, K.; von Heijne, G.; Uhlén, M.; Berglund, L. Prediction of the Human Membrane Proteome. Proteomics 2010 , 10, 1141–1149.
- Faller, R., MCB/BPH 241 Course Lectures. 2019.
- Gutierrez, M. G.; Mansfield, K. S.; Malmstadt, N. The Functional Activity of the Human Serotonin 5-HT1A Receptor Is Controlled by Lipid Bilayer Composition. Biophys. J. 2016 , 110 (11), 2486–2495.
- Hung, Y.;Schwarten, M.; Schünke, S.; Thiagarajan-Rosenkranz, P.; Hoffmann, S.; Sklan, E.; Koenig, B. (2015). Dengue virus NS4A cytoplasmic domain binding to liposomes is sensitive to membrane curvature. Biochimica Et Biophysica Acta (BBA) - Biomembranes, 1119-1126.
- Lee, A.G. (2004). How lipids affect the activities of integral membrane proteins. Biochimica Et Biophysica Acta (BBA) - Biomembranes, 62-87.
- Lee, A.G. (2011). Biological membranes: The importance of molecular detail. Trends in Biochemical Sciences, 493-500.
- Lee, A. G. Lipid-Protein Interactions. Biochem. Soc. Trans. 2011 , 39 (3), 761–766.
- Lehninger, A. (1982). Principles of biochemistry. New York, N.Y.: Worth.
- LeVine, M. V; Khelashvili, G.; Shi, L.; Quick, M.; Javitch, J. A.; Weinstein, H. Role of Annular Lipids in the Functional Properties of Leucine Transporter LeuT Proteomicelles. Biochemistry 2016 , 55 (6), 850–859
- McMahon, H. T.; Gallop, J. L. Membrane Curvature and Mechanisms of Dynamic Cell Membrane Remodelling. Nature 2005 , 438 (7068), 590–596.
- Phillips, R.; Ursell, T.; Wiggins, P.; Sens, P. Emerging Roles for Lipids in Shaping Membrane-Protein Function. Nature 2009 , 459, 379–385
- Razvan, L., & Thomas, D. (1994). Effects of Membrane Thickness on the Molecular Dynamics and Enzymatic Activity of Reconstituted Ca-ATPase. Biochemistry, 33, 2912-2920.
- Rholam, M., Scarlata, S., & Weber, G. (1984). Frictional resistance to the local rotations of fluorophores in proteins. Biochemistry, 6793-6796.
- Sezgin, E.; Levental, I.; Mayor, S.; Eggeling, C. The Mystery of Membrane Organization: Composition, Regulation and Roles of Lipid Rafts. Nat. Rev. Mol. Cell Biol. 2017 , 18 (6), 361–374
- Shinzawa-Itoh, K. et al. (2007) Structures and physiological roles of 13 integral lipids of bovine heart cytochrome c oxidase. EMBO J. 26, 1713– 1725
- Sperotto, M.M. A theoretical model for the association of amphiphilic transmembrane peptides in lipid bilayers. Eur. Biophys. J., 26 (1997), pp. 405–416
- Hydrophobic mismatch. (2013, September 20). In Wikipedia, The Free Encyclopedia. Retrieved 21:05, May 13, 2015, from en.Wikipedia.org/w/index.php?title=Hydrophobic_mismatch&oldid=5737297
- Engelman, D. M. Membranes Are More Mosaic than Fluid. Nature 2005 , 438 (7068), 578–580.
- van den Bogaart, G.; Meyenberg, K.; Risselada, H. J.; Amin, H.; Willig, K. I.; Hubrich, B. E.; Dier, M.; Hell, S. W.; Grubmüller, H.; Diederichsen, U.; et al. Membrane Protein Sequestering by Ionic Protein–Lipid Interactions. Nature 2011 , 479 (7374), 552–555.
- Palivan, C. G.; Goers, R.; Najer, A.; Zhang, X.; Car, A.; Meier, W. Bioinspired Polymer Vesicles and Membranes for Biological and Medical Applications. Chem. Soc. Rev 2016 , 45 , 377–411.
- Harayama, T.; Riezman, H. Understanding the Diversity of Membrane Lipid Composition. Nat. Rev. Mol. Cell Biol. 2018 , 19 , 281–296.
- Hasan, S. S.; Cramer, W. A. Internal Lipid Architecture of the Hetero-Oligomeric Cytochrome B6f Complex. Structure 2014 , 22 (7), 1008–1015.


13.6: Characteristics of Retailing Websites
- List the characteristics of an appealing retailer website
When a shopper enters a brick-and-mortar store, she reacts to the environment, the layout of the store and the product display to make her buying decision. If she’s online and lands on a website, she’s looking for a list of functions and qualities that are not unlike those we’ve already talked about . . . they’re built into the ecommerce platform to help the shoppers understand the type of product they’re going to get at this site, find what she’s looking for (and a whole lot more), and see examples of how she can use the products you offer.
An appealing retail website isn’t about looks, it’s about shoppability. Here are some of the qualities that make an ecommerce site shoppable.
- Ease of Use. This is the number one thing shoppers are looking for in an ecommerce site: they want it to be easy to use.
- Hi-res photos. Shoppers are looking for multiple views of an item in hi-res, so they can see every detail. And we should mention that those hi-res pictures should not take a long time to load. Adobe reports that 39% of consumers will drop off if the photos take too long to load. [1]
- Mobile-formatted site. An ecommerce site is more successful if it’s friendly to the mobile device user. In fact, Google lowered the boom on sites that aren’t mobile friendly in 2017, and they don’t get the same SEO (search engine optimization) considerations that mobile friendly sites do. That means that your site won’t come up near the top in a Google search.
- Free shipping. Is it worth it? Customers love free shipping so much that they’ll spend 30% more on average if free shipping is offered.
- User reviews. When shoppers buy a product they haven’t been able to touch and engage with, they want to hear how other users like it. Encouraging users to give reviews and featuring them prominently is important.
- Secure payment options. Not only should there be secure payment options, there should be cutting edge payment options. PayPal is passé . . . perhaps ACH, WePay and Skrill are options you want to offer. Customers are always looking for ways to keep their money and identity safe.
Yes, it should be eye-catching and visually attractive, too. For example, consider these two websites.
Which one is more inviting and easier to navigate?
These features listed above are what make the website “shoppable” and you can think of the webpage as the “store layout” of the ecommerce platform. Without good websites, your customer may not stay and shop.
Contributors and Attributions
- Characteristics of Retailing Websites. Authored by : Freedom Learning Group. Provided by : Lumen Learning. License : CC BY: Attribution
- Screenshot of Northernism's website. Provided by : Northernism. Located at : https://northernism.com/ . License : CC BY: Attribution
- Screenshot of arngren's website. Provided by : arngren. Located at : http://www.arngren.net/ . License : CC BY: Attribution


13.4: Grassland Biome
The Grassland biome is dominated by grasses of a variety of species, all having adapted to the summer drought common to their semiarid habitat. The broad expanse of the grasslands is occasionally broken by stands of trees. The midlatitude grasslands of have been exploited more than any other biome by humans. Ninety nine percent of the United State's tall grass prairie and 70 percent of the mixed grass and short grass in some states have disappeared. Eight-five percent of the state of Iowa was once covered in native prairie, only one percent is left. The grasslands, especially the tall grass prairies are the world's most productive agricultural areas. The famous corn belt of the United States was created on top of the rich brown mollisol soils that developed beneath the surface.
The Prairies
Prairies are found on the humid side of the grassland biome and are often referred to as the tall-grass prairie . A favorable annual moisture balance supports a dense ground cover of tall grasses. Grasses range in height between .6 to 1.2 meters (2 to 4 ft.), with some as tall as 8 feet or more on the eastern margin of the prairies in the United States. In the tall-grass prairie of Iowa, for example, typical grasses are big bluestem and little bluestem; a typical forb is black-eyed Susan. Trees and shrubs are generally limited to moist sites along stream channels or hill slopes facing away from the sun. The nutrient - rich soil beneath the grasslands drew farmers to these regions. Now, most natural tall-grass prairie has been replaced by agriculture. Extensive grasslands also occur in Argentina and the Ukraine.
The native grasslands of the world support a diversity of animal life. In North America, large grazers like the bison roamed the grasslands until hunted to near extinction by settlers moving west.
Steppe Grassland
On the drier side of the grassland biome lies the steppe grasslands . Vegetation must cope with the summer soil moisture deficit common to the steppe climate in which this formation class is found. Here, tall grass prairie gives way to grasses smaller than a half meter (2 ft).
Toward the drier portions the ground cover becomes sparse with patches of open ground found between clumps of grass. Overgrazing of the steppe vegetation leads to accelerated wind erosion and desertification.
Fires, especially those started by lightning, are a natural occurrence in the grassland biome. Fire destroys invasive species that compete with grasses. Fire suppression and farmland conversion have severely disrupted grassland ecosystems. Resource mangers now use prescribed burning to restore the health of prairie grasslands.
The Kalahari Desert actually isn't a desert under present conditions, though it is covered with much sand. It is a fossil desert found in the tropical steppe biome. Parts receive over 250 mm of precipitation, enough to support a cover of vegetation. It is a fossil desert found in the steppe biome. It' name is derived from the tribal word Khalagari, Kgalagadi or Kalagare meaning "a waterless place" or the Tswana word Keir, meaning "the great thirst".
Burrowing animals like ground squirrels, prairie dogs, pocket gophers are common in the steppe grasslands of North America. Burrowing predators like the black footed ferret are considered an endangered species.
Video : " National Bison Range Wildlife Refuge - near Missoula, Montana, MT."


10.6.2: Gender and Sex – Transgender and Intersex
-
- Last updated
- Save as PDF
- Miliann Kang, Donovan Lessard, Laura Heston, and Sonny Nordmarken
- University of Massachusetts Amherst via UMass Amherst Libraries
- Miliann Kang, Donovan Lessard, Laura Heston, and Sonny Nordmarken
- University of Massachusett Amherst
A binary gender perspective assumes that only men and women exist, obscuring gender diversity and erasing the existence of people who do not identify as men or women. A gendered assumption in our culture is that someone assigned female at birth will identify as a woman and that all women were assigned female at birth. While this is true for cisgender (or “cis”) individuals—people who identify in accordance with their gender assignment—it is not the case for everyone. Some people assigned male at birth identify as women, some people assigned female identify as men, and some people identify as neither women nor men. This illustrates the difference between, gender assignment , which doctors place on infants (and fetuses) based on the appearance of genitalia, and gender identity, which one discerns about oneself. The existence of transgender people , or individuals who do not identify with the gender they were assigned at birth, challenges the very idea of a single sex/gender identity. For example, trans women, women whose bodies were assigned male and who identify as women, show us that not all women are born with female-assigned bodies.  Transgender people may or may not have surgeries or hormone therapies to change their physical bodies, but in many cases they experience a change in their social gender identities. Some people who do not identify as men or women may identify as non-binary , gender fluid , or genderqueer, for example. Some may use gender-neutral pronouns, such as ze/hir or they/them, rather than the gendered pronouns she/her or he/his. As pronouns and gender identities are not visible on the body, trans communities have created procedures for communicating gender pronouns, which consists of verbally asking and stating one’s pronouns (Nordmarken, 2013).
The existence of sex variations fundamentally challenges the notion of a binary biological sex.  The bodies of individuals with sex characteristics variations do not fit typical definitions of what is culturally considered “male” or “female.” “Intersex,” like “female” and “male,” is a socially constructed category that humans have created to label bodies that they view as different from those they would classify as distinctly “female” or “male.” The term basically marks existing biological variation among bodies; bodies are not essentially intersex—we just call them intersex. The term is slightly misleading because it may suggest that people have complete sets of what would be called “male” and “female” reproductive systems, but those kinds of human bodies do not actually exist; “intersex” really just refers to biological variation. The term “hermaphrodite” is therefore inappropriate for referring to intersex, and it also is derogatory. There are a number of specific biological sex variations. For example, having one Y and more than one X chromosome is called Kleinfelter Syndrome.
Does the presence of more than one X mean that the XXY person is female? Does the presence of a Y mean that the XXY person is male?  Some people have genitalia that others consider ambiguous. This is not as uncommon as you might think. The Intersex Society of North America estimated that some 1.5% of people have sex variations—that is 2,000 births a year. So, why is this knowledge not commonly known? Many individuals born with genitalia not easily classified as “male” or “female” are subject to genital surgeries during infancy, childhood, and/or adulthood which aim to change this visible ambiguity. Surgeons reduce the size of the genitals of female-assigned infants they want to make look more typically “female” and less “masculine”; in infants with genital appendages smaller than 2.5 centimeters they reduce the size and assign them female (Dreger 1998). In each instance, surgeons literally construct and reconstruct individuals’ bodies to fit into the dominant, binary sex/gender system. While parents and doctors justify this practice as in “the best interest of the child,” many people experience these surgeries and their social treatment as traumatic, as they are typically performed without patients’ knowledge of their sex variation or consent. Individuals often discover their chromosomal makeup, surgical records, and/or intersex status in their medical records as adults, after years of physicians hiding this information from them. The surgeries do not necessarily make bodies appear “natural,” due to scar tissue and at times, disfigurement and/or medical problems and chronic infection. The surgeries can also result in psychological distress. In addition, many of these surgeries involve sterilization, which can be understood as part of eugenics projects, which aim to eliminate intersex people. Therefore, a great deal of shame, secrecy, and betrayal surround the surgeries. Intersex activists began organizing in North America in the 1990s to stop these nonconsensual surgical practices and to fight for patient-centered intersex health care. Broader international efforts emerged next, and Europe has seen more success than the first wave of mobilizations. In 2008, Christiane Völling of Germany was the first person in the world to successfully sue the surgeon who removed her internal reproductive organs without her knowledge or consent (International Commission of Jurists, 2008). In 2015, Malta became the first country to implement a law to make these kinds of surgeries illegal and protect people with sex variations as well as gender variations (Cabral & Eisfeld, 2015). Accord Alliance is the most prominent intersex focused organization in the U.S.; they offer information and recommendations to physicians and families, but they focus primarily on improving standards of care rather than advocating for legal change. Due to the efforts of intersex activists, the practice of performing surgeries on children is becoming less common in favor of waiting and allowing children to make their own decisions about their bodies. However, there is little research on how regularly nonconsensual surgeries are still performed in the U.S., and as Accord Alliance’s standards of care have yet to be fully implemented by a single institution, we can expect that the surgeries are still being performed.
The concepts of “transgender” and “intersex” are easy to confuse, but these terms refer to very different identities. To review, transgender people experience a social process of gender change, while intersex people have biological characteristics that do not fit with the dominant sex/gender system. One term refers to social gender (trans gender ) and one term refers to biological sex (inter sex ).  Gender theorists, such as Judith Butler and Gayle Rubin, have challenged the very notion that there is an underlying “sex” to a person, arguing that sex, too, is socially constructed. This is revealed in different definitions of “sex” throughout history in law and medicine—is sex composed of genitalia? Is it just genetic make-up? A combination of the two? Various social institutions, such as courts, have not come to a consistent or conclusive way to define sex, and the term “sex” has been differentially defined throughout the history of law in the United States. In this way, we can understand the biological designations of “male” and “female” as social constructions that reinforce the binary construction of men and women.


1.21: Dream Analysis Shows how Emotions are Difficult to Control
-
- Last updated
- Save as PDF
Dreams are significant because they reveal how the unconscious mind functions. In order to control your emotions it will help to understand how wild and crazy the unconscious mind is. The unconscious mind acts on impulses not rational thought. The experience of emotion is driven by the unconscious. When something happens you don’t think “I’m going to feel happy about that” you simply are happy for unconscious reasons. The unconscious mind works on impulses and learned responses. Since the “happy” response is learned it can be changed in part, however, because how your mind directs your emotions how to feel. It is important to understand, however, that emotions need to be lead on to how they should feel because they are illogical and function unconsciously, similar to how dreams function.
We need the escape of dreams from the logical, rational world in which we operate. There is a desire within humans to break everything down and tear everything apart. Why? Because breaking things is fun. No one wants to see everything continue as usual, why? Because things continuing as usual represents nothing out of the ordinary. Things that are out of the ordinary are going to be more emotional, and more stimulating. That’s why humans intentionally engineer their dreams, to have something fun to escape into. Take this dream “We’re in a hotel. We all have rooms, but we’re in Steve’s room. There are multiple beds that may be stacked. We are trying to make music. A boy starts playing guitar and it’s fantastic. Steve holds up my cell phone, it’s recording, he hands it to me. Steve asks me to play it back. There is a lot of music. One song my clarinet is so sharp. Steve says ‘if you can’t hear that…’ condescending. Steve leaves the room. We are competing for his attention, girls and boys. I am on a bed that is high. I know I’m the favorite and they’re asking me about it and I decide to leave. I slide off the bed, then reach up under the rail and grab a black candle (handmade) and a cigarette and something else.” It should be obvious that that is a fun event.
If you take all dreams and think about them, you will realize that they are fun, even nightmares are fun because they are emotional. It is fun for a person to have a deeply emotional experience because it is stimulating, people will do anything for stimulation even if that stimulation is a negative emotion. All dreams represent some sort of significant or large emotional event. The event doesn’t have to be real it just has to provoke a large emotional reaction in the person. As long as this emotional reaction doesn’t incur damage, then all emotional reactions are good. It is the saying, what doesn’t hurt you only makes you stronger, only it’s more like, what doesn’t hurt you only makes you stronger. So if it’s emotion, and it doesn’t hurt you, then it makes you stronger and you even like it.
 As solely an emotional being you aren’t engaging the logical part of your brain. So even if you dream about something like the death of your parent, you are still going to enjoy the dream because it is emotional and you’re not thinking about the consequences of that. That is why you dream, because dreaming is fun, even if it isn’t fun to think about when you wake up. If you were awake and thinking clearly you’d realize that you don’t want your parent to die, but during the dream you are solely and emotional being and just interested the thrill of the death of a loved one.
That is, you are interested in the emotional intensity of the death of a loved one because in dreams you are solely emotional. You are not thinking of the logical consequences, and therefore in dreams people are just emotional. There might be a little logic, but the emotional experience would tend to override it resulting in dreams like the death of relatives. The reason you might "enjoy" the death of a loved one is because the death causes you to think more about that person because you are emotionally involved in experiences such as deaths. While awake you are intellectually involved in experiences such as deaths and this intellectual involvement would lead to a realization that they are bad, but in dreams it would lead to no realization, just feeling for the person who is dying, which you might enjoy (not the fact that they are dying).
Why again would the death of a loved one be thrilling? Because it would be a huge emotional experience, and your system is interested in the shock of that experience, that is why you are likely to dream about it. In fact, any nightmare is just really a system shock that causes a healthy amount of anxiety. The person dreaming also “knows” that it is a dream when it is taking place. You know this because in dreams you don’t really worry about consequences, since they are just emotional to begin with. Logic means worrying and such, you can tell that if you had a dream of a death of a loved one, you wouldn’t worry about it in the dream, but you might worry about it while you are consciously awake. Let’s go back to the playing music in the hotel, if you are playing music in the hotel room, you aren’t going to worry about if there are other people near you in the dream that you might wake up (and you can tell that dreams are like that). But you are certainly going to think about it in reality. That’s because in dreams the emotional content is emphasized, and the dreaming mind isn’t aware that the logical one is going to be upset that the dream doesn’t make any sense when it wakes up, or that the logical one is going to be upset you killed a relative for fun.
Just because something is emotional doesn’t mean you worry about it while you are awake. Dreams try to eliminate thinking, the less thinking, the more emotional it is going to be. So dreams might have a lot of sexual content in them as well. You dream about things you want to experience, but only things you want to experience in the dreaming state. The dreaming state is a state in which you don’t have control over your body, and you have a very childish control over your emotions. Your emotions run free in dreams, if you want it, it’s yours (in the dream).  However, in the dream you aren’t really afraid because you aren’t clear thinking. It’s like why people like scary movies, it is something scary that you aren’t directly involved in, so you can safely experience it. You aren’t directly involved with the dream because it is a dream, it is not reality, and your mind responds to that by making dreams that are entertaining to watch, not to experience, so it is very similar to watching a movie, you’re equally distanced from the event.
It would be more real to watch something like a murder in real life then to watch a murder taking place in a dream, in the dream situation the murder might even seem fun. That is also how people can like watching violence in cartoons like Tom and Jerry, where all the characters do is beat each other up, people even find it amusing. Watching something like that of course in real life wouldn’t be amusing however (unless you’re sadistic). Dreams are just like cartoons, you’re not involved in it, it isn’t real, and if you are involved in the dream then it isn’t very physical since you can’t feel your limbs. You can even feel it, imagine a cartoon character in pain, is that fun or sad? It is fun because it is just the right amount of stimulation (it might be sad intellectually, but emotionally, like how dreams are emotional, it is fun). It’s the right amount of stimulation because your mind recognizes it as not real, you recognize logically that it is just a cartoon, or just a movie, and you don’t feel as bad as you would if it were real. That’s why in dreams we need more to properly stimulate us, because simply it isn’t real. That’s why dreams need to be more emotional and entertaining. If you had that much entertainment in real life (like if the dreams you had were actually real), you’d have way too much stimulation and you wouldn’t like it at all. Dreams just reflect the proper amount of stimulation you need to keep you stimulated. That’s probably why people dream at all, for the same reason people think all the time while they are awake, because boredom causes an incredible amount of anxiety. People simply need to think about something all of the time, even while they are asleep. But since it is a dream, they can think about things that aren’t realistic and don’t make sense so they can have fun during those dreams. Doing something like moving some stuff around might be entertaining in real life because you are physically doing it, but in a dream it just wouldn’t suffice, you would need something spicy taking place like death, sex, fear, desire, emotion, or strong emotion.
Dreams in general tend to be weird.  Things in dreams often don’t make any sense in reality, but dreams are often incredibly sophisticated at the same time. This would suggest that dreams are emotional, not logical. Emotion is very complicated, but it often doesn’t make any logical sense. Dreams convey feelings very well, they amplify feelings, they don’t amplify logic.
For example, say you were thinking about a toothbrush that day, or had a lot of thoughts about brushing your teeth, or had some trouble with the dentist and it was bothering you. In your dream that night, you wouldn’t think about the events of the day, or logically think about how you could fix your tooth problem. In fact the logical thing would probably never occur in your dream, that would be out of character since dreams are more emotional, you’d probably never dream thinking “ah I should brush my teeth more thoroughly”.   See how one is more emotional than the other?
 This is easy to prove, think of any dream you’ve ever had, or ever heard of, whatever it was, it didn’t make complete sense.  That makes sense since if you were actually thinking, you’d want to experience real emotions and move your body around to get that experience, not just think about them.
This might make dreams more sexual or Freudian, but more importantly anything that is most strongly emotional to the person having the dream. Take this dream for example “I was at a type of arena-ish thing but it had balconies like a theater would.” Notice first off that it doesn’t make sense, arena’s don’t have balconies like a theater would. Clearly if the person was thinking clearly she/he wouldn’t have been able to put theater balconies in an arena. Now there sometimes are balconies in an arena, but this person must have been referring to balconies that were pretty like they are in theaters with strong contrast to the arena, say like a stone arena with pretty wooden balconies in pink and stuff in them. That description I gave sounds like a typical dream because it doesn’t make sense, and due to the contrast/mix of the arena and the theater, it is very emotional.
The mix of the two things makes it more emotional because it is something which you wouldn’t find anywhere in reality. Things that stand out tend to be more emotional, and anything that doesn’t make sense, like doesn’t make ANY sense, is going to be emotional because it stands out from your everyday experience. Something like a giant gumball rolling over and over in your head, that doesn’t make any sense, and its emotional. But why is it emotional? It is because you never find giant gumballs (that are chewed just standing around outside) so if you found one, you’d be in shock, and very emotional.
There are things that are emotional and can be found in real life of course.  It would probably be more emotional for the dreamer to be doing something with Mel Gibson, since it’s not likely he’ll ever do something with Mel and therefore would find it rare when he did, so it’s a not realistic, out of the ordinary, emotional experience. Furthermore they are using gold swords, how often are gold swords used? Gold is a more emotional color than steel as well. Color is emotional, so color, a dramatic color, or large color contrasts are often found in dreams to further amplify emotion.
Take this dream, see how emotional it is, emotional, not realistic, and amplified for dramatic content.
“I am the best student in a hard science class of some sort. Every day before class I hold study sessions. Everyone fails the first test but me. We are all milling about in the hall after class. The teacher and some other students express interest in the study sessions, but I say I don't really need them. They seem disappointed. Then I tell everyone "Hey, all those study sessions that I've been having... BY MYSELF... will still be there next week" inviting them. The professor asks anyone with a disease to hang around and see her in ten minutes, saying she has the shakes. She's very concerned with her health, which has been strange for some time. I think about staying, but I leave. I see Joe Horvath in the hall and hug him, but I see that he has a finger the looks like it was smashed and healed flattish and deformed. There are flecks of blue paint or nail polish or the nail is flecked blue. When I ask him about it he says he didn't even notice and doesn't know what happened, but it doesn't hurt.”
The dreamer thinks he is the best in the class, not just any class, but a hard science class. He is so much better than anyone else, that he has “study sessions” by himself. Of course that doesn’t make any sense, the people were asking him about a study session, implying that a study session would involve more than one person, like they usually do. But in his dream he forgets logic and all of a sudden he is the only person needed for a study session, in real life he wouldn’t have said that because it just wouldn’t be a proper thing to say - he wouldn't say something that silly in real life. To make the dream even more emotional another out of the ordinary event is occurring: the teacher is feeling sick, and her health has been “strange for some time” not bad for sometime, but strange for some time, the word strange would imply something really out of the ordinary going on, like an extraterrestrial disease or something weird, the weirdness and out of the ordinariness being added for extra emotional content, of course. Does this mean that the dreamer is afraid of a strange disease? No it just means he is trying to entertain himself in his sleep by adding extra dramatic content by using the word strange, instead of bad. (it’s extremely rare to use the word strange when describing that one is sick, so what I suggested about extraterrestrial implications makes more sense). When you say, “oh I’ve been feeling strange lately” you are implying that something really weird is going on with you (or in this case your health) which would bring up further rise for concern, or a further rise in emotional, dramatic content!
Take this dream “We’re in a hotel. We all have rooms, but we’re in Steve’s room. There are multiple beds that may be stacked. We are trying to make music. A boy starts playing guitar and it’s fantastic. Steve holds up my cell phone, it’s recording, he hands it to me. Steve asks me to play it back. There is a lot of music. One song my clarinet is so sharp. Steve says ‘if you can’t hear that…’ condescending. Steve leaves the room. We are competing for his attention, girls and boys. I am on a bed that is high. I know I’m the favorite and they’re asking me about it and I decide to leave. I slide off the bed, then reach up under the rail and grab a black candle (handmade) and a cigarette and something else.” That is also very out of the ordinary, in fact that would probably never actually happen in real life because everyone in the hotel would hear the music. The dreamer obviously wasn’t logically, clearly thinking. If she/he was then the dream would have ended with the people next door complaining about the noise, or there being somewhere in the dream something about checking to see if the hall was clear, but even then someone might walk down it. The point is it is very out of the ordinary, which, since it is rare, is probably more emotional solely because it’s a new and exciting experience that you furthermore can’t have in real life, so it also has that “I want it since I can’t have it” emotional feel. This is the real kicker, you can sense that the dream wouldn’t have made any sense if they actually checked to see if there were other people in the hall. It is only an ordinary, regular dream, if it doesn’t make sense. And you can sense that that is true.
Let’s see how out of the ordinary this dream is. (All this so far proves that dreams are out of the ordinary, probably just to add emotional content because of the contrast with reality). “We are rehearsing. Instead of a lyrics sheet there is a flat piece of 3D art. It’s a series of concentric circles.  That’s the verse I am supposed to sing. I get singled out and have to sing the verse alone. It’s about life going around and down forever. There’s an infinity symbol.”
For starters there is no such thing as a flat piece of 3D art, 3D is 3D, but you can see how that would be fun for the dreamer to think about, entertaining for him to think about how it could be 3D, yet not 3D at the same time. This emphasizes the emotional content, but it low on the logical content. Why is the emotional content emphasized? Because dreams are for entertainment, you’re trying to have fun in your dream. So he/she mixes the lyrics sheet, 3D art, and flat together. That’s a fun thing to do. Dreams in general are going to be more on the fun side, less on the logical, ah this makes sense side.  That just doesn’t make any sense. Exactly, that’s what is fun about it, trying to imagine something that doesn’t make any sense. Trying to put together in reality, things that just can’t be put together. It’s like you’re trying and trying to do something that just can’t be done.  It’s fun to try and break reality and put things together that don’t belong together. That way you create something new and different, something you’d want to dream about. People don’t want to think clearly in dreams, they want to relax, have fun, and do things that they never could in reality. See things they’ve never seen, and experience emotions that they aren’t going to be able to experience in other places.


2.2: Systems of Linear Equations and the Gauss-Jordan Method
-
- Last updated
- Save as PDF
Learning Objectives
In this section you will learn to
- Represent a system of linear equations as an augmented matrix
- Solve the system using elementary row operations.
In this section, we learn to solve systems of linear equations using a process called the Gauss-Jordan method. The process begins by first expressing the system as a matrix, and then reducing it to an equivalent system by simple row operations. The process is continued until the solution is obvious from the matrix. The matrix that represents the system is called the augmented matrix , and the arithmetic manipulation that is used to move from a system to a reduced equivalent system is called a row operation .
Example \(\PageIndex{1}\)
Write the following system as an augmented matrix.
\[\begin{array}{l}
2 x+3 y-4 z=5 \\
3 x+4 y-5 z=-6 \\
4 x+5 y-6 z=7
\end{array}\nonumber \]
Solution
We express the above information in matrix form. Since a system is entirely determined by its coefficient matrix and by its matrix of constant terms, the augmented matrix will include only the coefficient matrix and the constant matrix. So the augmented matrix we get is as follows:
\[\left[\begin{array}{ccc|c}
2 & 3 & -4 & 5 \\
3 & 4 & -5 & -6 \\
4 & 5 & -6 & 7
\end{array}\nonumber\right]\nonumber \]
In the last section, we expressed the system of equations as \(AX = B\), where \(A\) represented the coefficient matrix, and \(B\) the matrix of constant terms. As an augmented matrix, we write the matrix as \(\left[\begin{array}{l|l}A & B \end{array}\right]\). It is clear that all of the information is maintained in this matrix form, and only the letters \(x\), \(y\) and \(z\) are missing. A student may choose to write \(x\), \(y\) and \(z\) on top of the first three columns to help ease the transition.
Example \(\PageIndex{2}\)
For the following augmented matrix, write the system of equations it represents.
\[\left[\begin{array}{ccccc}
1 & 3 & -5 & | & 2 \\
2 & 0 & -3 & | & -5 \\
3 & 2 & -3 & | & -1
\end{array}\right]\nonumber \]
Solution
The system is readily obtained as below.
\[\begin{array}{l}
x+3 y-5 z=2 \\
2 x-3 z=-5 \\
3 x+2 y-3 z=-1
\end{array}\nonumber \]
Once a system is expressed as an augmented matrix, the Gauss-Jordan method reduces the system into a series of equivalent systems by using the row operations. This row reduction continues until the system is expressed in what is called the reduced row echelon form . The reduced row echelon form of the coefficient matrix has 1's along the main diagonal and zeros elsewhere. The solution is readily obtained from this form.
The method is not much different form the algebraic operations we employed in the elimination method in the first chapter. The basic difference is that it is algorithmic in nature, and, therefore, can easily be programmed on a computer.
We will next solve a system of two equations with two unknowns, using the elimination method, and then show that the method is analogous to the Gauss-Jordan method.
Example \(\PageIndex{3}\)
Solve the following system by the elimination method.
\[\begin{array}{l}
x+3 y=7 \\
3 x+4 y=11
\end{array} \nonumber \]
Solution
We multiply the first equation by – 3, and add it to the second equation.
\begin{aligned}
-3 x-9 y &=-21 \\
3 x+4 y &=11 \\ \hline
-5y &=-10
\end{aligned}
By doing this we transformed our original system into an equivalent system:
\begin{aligned}
x+3 y &=7 \\
-5 y &=-10
\end{aligned}
We divide the second equation by – 5, and we get the next equivalent system.
\begin{aligned}
x+3 y &=7 \\
y &=2
\end{aligned}
Now we multiply the second equation by – 3 and add to the first, we get
\[\begin{array}{l}
x=1 \\
y=2
\end{array}\nonumber \]
Example \(\PageIndex{4}\)
Solve the following system from Example 3 by the Gauss-Jordan method, and show the similarities in both methods by writing the equations next to the matrices.
\begin{array}{l}
x+3 y=7 \\
3 x+4 y=11
\end{array}
Solution
The augmented matrix for the system is as follows.
\[\left[\begin{array}{cccc}
1 & 3 & | & 7 \\
3 & 4 & | & 11
\end{array}\right] \quad\left[\begin{array}{c}
x+3 y=7 \\
3 x+4 y=11
\end{array}\right] \nonumber \]
We multiply the first row by – 3, and add to the second row.
\[\left[\begin{array}{cccc}
1 & 3 & | & 7 \\
0 & -5 & | & -10
\end{array}\right] \quad\left[\begin{array}{c}
x+3 y&=7 \\
-5 y&=-10
\end{array}\right] \nonumber \]
We divide the second row by – 5, we get,
\[\left[\begin{array}{llll}
1 & 3 & | & 7 \\
0 & 1 & | & 2
\end{array}\right] \quad\left[\begin{array}{rl}
x+3 y & =7 \\
y & =2
\end{array}\right] \nonumber \]
Finally, we multiply the second row by – 3 and add to the first row, and we get,
\[\left[\begin{array}{llll}
1 & 0 & | & 1 \\
0 & 1 & | & 2
\end{array}\right] \quad\left[\begin{array}{l}
x=1 \\
y=2
\end{array}\right] \nonumber \]
Now we list the three row operations the Gauss-Jordan method employs.
Row Operations
- Any two rows in the augmented matrix may be interchanged.
- Any row may be multiplied by a non-zero constant.
- A constant multiple of a row may be added to another row.
One can easily see that these three row operation may make the system look different, but they do not change the solution of the system.
The first row operation states that if any two rows of a system are interchanged, the new system obtained has the same solution as the old one. Let us look at an example in two equations with two unknowns. Consider the system
\begin{aligned}
x+3 y&=7 \\
3 x+4 y&=11
\end{aligned}
We interchange the rows, and we get,
\begin{aligned}
3 x+4 y&=11 \\
x+3 y&=7
\end{aligned}
Clearly, this system has the same solution as the one above.
The second operation states that if a row is multiplied by any non-zero constant, the new system obtained has the same solution as the old one. Consider the above system again,
\begin{aligned}
x+3 y&=7 \\
3 x+4 y&=11
\end{aligned}
We multiply the first row by –3, we get,
\begin{aligned}
-3 x-9 y &=-21 \\
3 x+4 y &=11
\end{aligned}
Again, it is obvious that this new system has the same solution as the original.
The third row operation states that any constant multiple of one row added to another preserves the solution. Consider our system,
\begin{aligned}
x+3 y&=7 \\
3 x+4 y&=11
\end{aligned}
If we multiply the first row by –3, and add it to the second row, we get,
\begin{aligned}
x+3 y&=7 \\
-5 y&=-10
\end{aligned}
And once again, the same solution is maintained.
Now that we understand how the three row operations work, it is time to introduce the Gauss-Jordan method to solve systems of linear equations. As mentioned earlier, the Gauss-Jordan method starts out with an augmented matrix, and by a series of row operations ends up with a matrix that is in the reduced row echelon form .
A matrix is in the reduced row echelon form if the first nonzero entry in each row is a 1, and the columns containing these 1's have all other entries as zeros. The reduced row echelon form also requires that the leading entry in each row be to the right of the leading entry in the row above it, and the rows containing all zeros be moved down to the bottom. We state the Gauss-Jordan method as follows.
Gauss-Jordan Method
- Write the augmented matrix.
- Interchange rows if necessary to obtain a non-zero number in the first row, first column.
- Use a row operation to get a 1 as the entry in the first row and first column.
- Use row operations to make all other entries as zeros in column one.
- Interchange rows if necessary to obtain a nonzero number in the second row, second column. Use a row operation to make this entry 1. Use row operations to make all other entries as zeros in column two.
- Repeat step 5 for row 3, column 3. Continue moving along the main diagonal until you reach the last row, or until the number is zero.
The final matrix is called the reduced row-echelon form.
Example \(\PageIndex{5}\)
Solve the following system by the Gauss-Jordan method.
\begin{aligned}
2 x+y+2 z &=10 \\
x+2 y+z &=8 \\
3 x+y-z &=2
\end{aligned}
Solution
We write the augmented matrix.
\[\left[\begin{array}{ccccc}
2 & 1 & 2 & | & 10 \\
1 & 2 & 1 & | & 8 \\
3 & 1 & -1 & | & 2
\end{array}\right] \nonumber \]
We want a 1 in row one, column one. This can be obtained by dividing the first row by 2, or interchanging the second row with the first. Interchanging the rows is a better choice because that way we avoid fractions.
\[\left[\begin{array}{ccccc}
1 & 2 & 1 & | & 8 \\
2 & 1 & 2 & | & 10 \\
3 & 1 & -1 & | & 2
\end{array}\right] \quad \text { we interchanged row 1 (R1) and row 2 (R2) }\nonumber \]
We need to make all other entries zeros in column 1. To make the entry (2) a zero in row 2, column 1, we multiply row 1 by - 2 and add it to the second row. We get,
\[\left[\begin{array}{ccccc}
1 & 2 & 1 & | & 8 \\
0 & -3 & 0 & | & -6 \\
3 & 1 & -1 & | & 2
\end{array}\right] \quad-2 R 1+R 2 \nonumber \]
To make the entry (3) a zero in row 3, column 1, we multiply row 1 by - 3 and add it to the third row. We get,
\[\left[\begin{array}{ccccc}
1 & 2 & 1 & | & 8 \\
0 & -3 & 0 & | & -6 \\
0 & -5 & -4 & | & -22
\end{array}\right] \quad-3 R 1+R 3 \nonumber \]
So far we have made a 1 in the left corner and all other entries zeros in that column. Now we move to the next diagonal entry, row 2, column 2. We need to make this entry(–3) a 1 and make all other entries in this column zeros. To make row 2, column 2 entry a 1, we divide the entire second row by –3.
\[\left[\begin{array}{ccccc}
1 & 2 & 1 & | & 8 \\
0 & 1 & 0 & | & 2 \\
0 & -5 & -4 & | & -22
\end{array}\right] \quad \mathrm{R} 2 \div(-3) \nonumber \]
Next, we make all other entries zeros in the second column.
\[\left[\begin{array}{ccccc}
1 & 0 & 1 & | & 4 \\
0 & 1 & 0 & | & 2 \\
0 & 0 & -4 & | & -12
\end{array}\right] \quad-2 R 2+R 1 \text { and } 5 R 2+R 3 \nonumber \]
We make the last diagonal entry a 1, by dividing row 3 by – 4.
\[\left[\begin{array}{ccccc}
1 & 0 & 1 & | & 4 \\
0 & 1 & 0 & | & 2 \\
0 & 0 & 1 & | & 3
\end{array}\right] \quad \quad R 3 \div(-4) \nonumber \]
Finally, we make all other entries zeros in column 3.
\[\left[\begin{array}{ccccc}
1 & 0 & 0 & | & 1 \\
0 & 1 & 0 & | & 2 \\
0 & 0 & 1 & | & 3
\end{array}\right] \quad-\mathrm{R} 3+\mathrm{R} 1 \nonumber \]
Clearly, the solution reads \(x =1\), \(y = 2\), and \(z = 3\).
Before we leave this section, we mention some terms we may need in the fourth chapter.
The process of obtaining a 1 in a location, and then making all other entries zeros in that column, is called pivoting .
The number that is made a 1 is called the pivot element , and the row that contains the pivot element is called the pivot row .
We often multiply the pivot row by a number and add it to another row to obtain a zero in the latter. The row to which a multiple of pivot row is added is called the target row .