11: The Interaction of Selection, Mutation, and Migration
The Interaction of Selection, Mutation, and Migration.
Genetic variation is the raw fuel of evolution. Without variation, natural selection would have nothing to act on to shape adaptive traits. However, variation can be deleterious.
Mutation, broadly defined, is the ultimate source of all genetic variation and is constantly introducing new variation into all populations. However, mutation is random and so mutations that affect function are often damaging. Thus mutation will, in the absence of sufficiently strong selection, degrade pre-existing adaptations and undo the work of selection that has built up functional regions of DNA over time.
Migration, the movement of individuals into a population, can also increase variation to the population as the individuals bring new alleles in from surrounding populations. Thus migration can be an important source of adaptive alleles, aiding their spread amongst populations within a species. Adaptive alleles can even spread between species if low levels of interbreeding occur. However, again, just like mutation, migration can disrupt adaptations. When populations are locally adapted migration amongst populations can introduce maladaptive alleles into well adapted populations. If this migration pressure is sufficiently strong, it can lead to the collapse of local adaptations, or even the collapse of species.
In this chapter we’ll study some of the interplay between selection, migration, and mutation.
Mutation–Selection Balance
Mutation is constantly introducing new alleles into the population. Therefore, variation can be maintained within a population not only if selection is balancing (e.g. through heterozygote advantage or fluctuating selection over time, as we have seen in the previous section), but also due to a balance between mutation introducing deleterious alleles and selection acting to purge these alleles from the population . To study mutation-selection balance, we return to the model of directional selection, where allele \(A_1\) is advantageous, i.e.
| genotype | \(A_1A_1\) | \(A_1A_2\) | \(A_2A_2\) |
| absolute fitness | \(W_{11}\) | \(\geq W_{12} \geq\) | \(W_{22}\) |
| relative fitness | \(w_{11}=1\) | \(w_{12}=1-sh\) | \(w_{22}=1-s\) . |
We’ll begin by considering the case where allele \(A_2\) is not completely recessive ( \(h>0\) ), so that the heterozygotes suffer at least some disadvantage. We denote by \(\mu = \mu_{1\rightarrow2}\) the mutation rate per generation from \(A_1\) to the deleterious allele \(A_2\) , and assume that there is no reverse mutation ( \(\mu_{2\rightarrow1} = 0\) ). Let us assume that selection against \(A_2\) is relatively strong compared to the mutation rate, so that it is justified to assume that \(A_2\) is always rare, i.e. \(q_t = 1-p_t \ll 1\) . Compared to previous sections, for mathematical clarity, we also switch from following the frequency \(p_t\) of \(A_1\) to following the frequency \(q_t\) of \(A_2\) . Of course, this is without loss of generality. The change in frequency of \(A_2\) due to selection can be written as
\[\Delta_S q_t = \frac{ \overline{w}_2 - \overline{w}_1}{\overline{w}} p_t q_t \approx -hs q_t. \label{eq:dirSelApprox}\]
This approximation can be found by assuming that \(q^2 \approx 0\) , \(p \approx 1\) , and that \(\overline{w} \approx w_1\) .
It is worth pointing out that the fitness of the \(A_2A_2\) homozygote has not entered this calculation, as \(A_2\) is so rare that it is hardly ever found in the homozygous state. Therefore, if \(A_2\) has any deleterious effect in a heterozygous state (i.e. if \(h>0\) ), it is this effect that determines the frequency at which \(A_2\) is maintained in the population. Also, note that by writing the total change in allele frequency as \(\Delta_M q_t + \Delta_S q_t\) we have implicitly assumed that we can ignore terms of order \(\mu \times s\) . That is, we have assumed that mutation and selection are both relatively weak. This assumption is valid under our prior assumption that both \(\mu\) and \(s\) are small.
If an allele is truly recessive (although few likely are), we have \(h=0\) , and so Equation [eqn:mut_sel_bal] is not valid. However, we can make an argument similar to the one above to show that, for truly recessive alleles,
\[q_e = \sqrt{\frac{\mu}{s}}. \label{eqn:recess_mut_sel_bal}\]
Oblong-winged katydids ( Amblycorypha oblongifolia ) are usually green. However, some are bright pink, thanks to an erythrism mutation . This pink condition is thought to be due to a dominant mutation ( Crew, 2013 ). Assume that roughly one in ten thousand katydids is bright pink and that the mutation rate at the gene underlying this condition is \(10^{-5}\) . What is the relative fitness of heterozygotes for the pink mutation?
The genetic load of deleterious alleles
What effect do such deleterious mutations at mutation–selection balance have on the population? It is common to quantify the effect of deleterious alleles in terms of a reduction of the mean relative fitness of the population. For a single site at which a deleterious mutation is segregating at frequency \(q_e = \mu/(hs)\) , the population mean relative fitness is reduced to
\[\overline{w} = 1- 2p_e q_e hs - q_e^2s \approx 1-2\mu. \label{eqn:mut_load}\]
Somewhat remarkably, the drop in mean fitness due to a site segregating at mutation–selection balance is independent of the selection coefficient against the heterozygote; it depends only on the mutation rate . Intuitively this is because, given a fixed mutation rate, less deleterious alleles can rise to a higher equilibrium frequency, and thus contribute the same total load as more deleterious (rarer) alleles, but this load is spread across more individuals in the population. Note that this result applies only if the mutation is not totally recessive, i.e. if \(h > 0\) .
A fitness reduction of \(2\mu\) is very small, given that the mutation rate of a gene is usually on the order of \(10^{-5}\) . However, if there are many loci segregating at mutation–selection balance, small fitness reductions can accumulate to a substantial so-called genetic load, a major cause of variation in fitness-related traits among individuals. To see how all of these loci contribute to variation in fitness consider the fact that the human genome contains over twenty thousand protein-coding genes, and many other functional regions, the vast majority of which will be subject to purifying selection against mutations that disrupt their function. In humans, most loss of function (LOF) variants, which severely disrupt a protein-coding gene, are found at low frequencies but each human genome typically carries over a hundred LOF variants . Not every LOF allele will be deleterious; some could even be advantageous. However, the combined load of these LOF alleles must on average lower our fitness, otherwise selection wouldn’t be removing them from the population. Each one of us carries a unique set of these LOF alleles, usually in a heterozygous state. We differ slightly in how many of these alleles we carry. For example, the left side of Figure \(\PageIndex{1}\) shows the distribution of the number of LOF alleles carried by 769 individuals of Dutch ancestry. The individuals who carry fewer of these LOF alleles will on average likely have higher fitness than those individuals with more. We don’t yet know how much fitness variation this explains across individuals, nor do we know how most of these LOF alleles manifest their fitness consequences through disease and other mechanisms. However, it’s a reasonable guess that this variation in LOF alleles, presumably maintained by mutation-selection balance, is a major source of variation in fitness.
Inbreeding depression
All else being equal, Equation [eqn:mut_sel_bal] suggests that mutations that have a smaller effect in the heterozygote can segregate at higher frequency under mutation–selection balance. As a consequence, alleles that have strongly deleterious effects in the homozygous state can still segregate at low frequencies in the population, as long as they do not have too strong a deleterious effect in heterozygotes. Thus, outbred populations may have many alleles with recessive deleterious effects segregating within them.
Assume that a deleterious allele has a relative fitness \(0.99\) in heterozygotes and a relative fitness \(0.2\) when present in the homozygote state. Assume that the deleterious allele is at a frequency \(10^{-3}\) at birth and the genotype frequencies follow from HWE. Only considering the fitness effects of this locus, and measuring fitness relative to the most fit genotype, answer the following questions:
A) What is the average fitness of an individual in the population?
B) What is the average fitness of the child of a full-sib mating?
One consequence of segregating for low-frequency recessive deleterious alleles is that inbreeding can reduce fitness. In typically outbred populations, the mean fitness of individuals decreases with the inbreeding coefficient, i.e. so-called ’inbreeding depression’ is a common observation. This wide-spread observation dates back to systematic surveys of inbreeding depression by . Inbreeding depression is likely primarily a consequence of being homozygous at many loci for alleles with recessive deleterious effects.
One example of inbreeding depression is shown in Figure \(\PageIndex{3}\). White campion ( Silene latifolia ) is a dioecious flowering plant; dioecious means that the males and females are separate individuals. performed crosses to create offspring who were outbred, the offspring of half-sibs, full-sibs, and of two generations of full-sib mating. He measured their germination success, which is plotted in Figure \(\PageIndex{3}\). Note how the fitness of individuals declines with increased inbreeding.
We also see evidence for inbreeding depression in various human populations. For example, used the remarkable genealogical records in Iceland to look at the effects of inbreeding on various fitness components in humans. They saw that parents who were closer than 2nd cousins had children with reduced lifespans. However, these patterns were more complex for other fitness components with parents with immediate levels of relatedness having more descendants overall. More generally, studying inbreeding depression is challenging in humans because it can be difficult to differentiate the cultural and socio-economic effects frombiological effects on reproduction. Finally, it is important to note that societal views of inbreeding risks can be overblown compared to the actual risks and that these fears have often been used to stigmatize immigrant and rural poor communities .
Purging the inbreeding load.
Populations that regularly inbreed over sustained periods of time are expected to partially purge this load of deleterious alleles. This is because such populations have exposed many of these alleles in a homozygous state, and so selection can more readily remove these alleles from the population.
If the population has sustained inbreeding, such that individuals in the population have an inbreeding coefficient \(F\) , deleterious alleles at each locus will find a new equilibrium frequency. Assuming the mutation-selection model, now with inbreeding, the equilibrium frequency is
\[q_e = \frac{\mu}{\big( h(1-F) + F \big) s}\]
The frequency of the deleterious allele is decreased due to the allele now being expressed in homozygotes, and therefore exposed to selection, more often due to inbreeding. Thus, all else being equal, populations that have had a long-term history of close inbreeding will purge their load.
Migration–selection balance
The influx of alleles carried by migrants from other populations can be an important source of genetic and phenotypic variation. Thus, one reason for the persistence of deleterious alleles in a population is that there is a constant influx of maladaptive alleles from other populations where these alleles are locally adaptive. Migration–selection balance seems unlikely to be as broad an explanation for the persistence of deleterious alleles genome-wide as mutation-selection balance. However, a brief discussion of such alleles is worthwhile, as it helps to inform our ideas about local adaptation, hybrid zones, and speciation.
Local adaptation can occur over a range of geographic scales. Local adaptation is relatively unimpeded by migration at broad geographically scales, where selection pressures change more slowly than distances over which individuals typically migrate over a number of generations. Adaptation can, however, potentially occur on much finer geographic scales, from kilometers down to meters in some species. On such small scales, dispersal is surely rapidly moving alleles between environments, but local adaptation is maintained by the continued action of selection. An example of adaptation at fine-scales is shown in Figure \(\PageIndex{7}\) . studied the patterns of heavy-metal resistance in plants on mine tailings and in nearby meadows, a set of classic studies of population differences maintained by local adaptation to different soils.
Even at these very short geographically scales, over which seed and pollen will definitely move, we see strong local adaptation. Zinc-intolerant alleles are nearly absent from the mine tailings because they prevent plants from growing on these zinc-heavy soils; conversely, zinc-tolerant alleles do not spread into the meadow populations, likely due to some trade-off or fitness cost of zinc-tolerance.
As a first pass at developing a model of local adaptation, let’s consider a haploid two-allele model with two different populations, see Figure \(\PageIndex{8}\), where the relative fitnesses of our alleles are as follows
| allele | \(1\) | \(2\) |
|---|---|---|
| population 1 | 1 | 1-s |
| population 2 | 1-s | 1 |
As a simple model of migration, let’s suppose within a population a fraction of \(m\) individuals are migrants from the other population, and \(1-m\) individuals are from the same population.
To quickly sketch an equilibrium solution to this scenario, we’ll take an approach analogous to our mutation-selection balance model. To do this, let’s assume that selection is strong compared to migration ( \(s \gg m\) ), such that allele \(1\) will be almost fixed in population \(1\) and allele \(2\) will be almost fixed in population \(2\) . If that is the case, migration changes the frequency of allele \(2\) in population \(1\) ( \(q_1\) ) by
\[\Delta_{Mig.} q_1 \approx m\]
while as noted above \(\Delta_{S} q_1= -sq_1\) , so that migration and selection are at an equilibrium when \(0 = \Delta_{S} q_1+ \Delta_{Mig.}q_1\) , i.e. an equilibrium frequency of allele \(2\) in population \(1\) of
\[q_{e,1} = \frac{m}{s}\]
Here, migration is playing the role of mutation and so migration–selection balance (at least under strong selection) is analogous to mutation–selection balance.
We can use this same model by analogy for the case of migration–selection balance in a diploid model. For the diploid case, we replace our haploid \(s\) by the cost to heterozygotes \(hs\) from our directional selection model, resulting in a diploid migration–selection balance equilibrium frequency of
\[q_{e,1} = \frac{m}{hs} \label{eqn:mig_sel_eq}\]
If selection is weaker and only of the order of migration \(s \approx m\) our migration-selection polymorphism collapses, as selection can not maintain the difference in the face of gene flow. Under this situation, both populations are expected to have roughly the same frequency of the alleles. Migration has swamped out local adaptation.
HOEKSTAR et al. (2004) found that the dark \(D\) allele was at \(3\%\) frequency at the Tule Mountains study site. Using \(F_{ST}\) -based approaches, for unlinked markers, they estimated that the per individual migration rate was \(m=7.0 \times 10^{-4}\) per generation between this site and the Pinacate lava flow. What is the selection coefficient acting against the dark \(D\) allele at the Tule Mountains site?
As an example of fine-scale local adaptation due to a single locus, consider the case of the rock pocket mice adapting to lava flows. Throughout the deserts of the American Southwest there are old lava flows, where the rocks and soils are much dark than the surrounding desert. Many populations of small animals that live on these flows have evolved darker pigmentation to be cryptic against this dark substrate and better avoid visual predators. One example of such a locally adapted population are the rock pocket mice ( Chaetodipus intermedius ) who live on the Pinacate lava flow on the Arizona-Mexico border, studied by . These mice have much darker, more melanic pelts than the mice who live on nearby rocky outcrops (see Figure \(\PageIndex{9}\)). determined that a dominant allele ( \(D\) ) at MC1R is the primary determinant of this melanic phenotype. The frequency of this allele across study sites is shown in Figure \(\PageIndex{9}\). found that other, unlinked markers showed little differentiation over these populations, suggesting that the migration rate is high.
The width of a genetic cline.
We can also extend these ideas beyond our discrete model to a model of a population spread out on a landscape where individuals migrate in a more continuous fashion. For simplicity, let’s assume a one dimensional habitat, where the habitat makes a sharp transition in the middle of our region. You could imagine this to be a set of populations sampled along a transect through some environmental transition. Our individuals disperse to live on average \(\sigma\) miles away from where they were born (we can think of this as our individuals migrating a random displacement drawn from a normal distribution, with mean zero, and \(\sigma\) being the standard deviation of this distribution). We’ll think of a bi-allelic model where the homozygotes for allele 1 have an additive selective advantage \(s\) over allele \(2\) homozygotes to the east of our habitat transition (left of zero in Figure \(\PageIndex{11}\)). This flips to allele \(2\) having the same advantage \(s\) west of the transition (right of zero). If you’ve read this send Prof Coop a picture of the East and West Beast.
With this setup, we get an equilibrium distribution of our two alleles, where to the left of zero our allele \(2\) is at higher frequency, while to the right of zero allele \(1\) predominates. As we cross from the left to the right side of our range, the frequency of our allele \(2\) decreases in a smooth cline. The frequency of allele \(2\) , \(q(x)\) , is shown as a function of location, \(x\) , along the cline for a variety of selection coefficients ( \(s\) ) in Figure \(\PageIndex{11}\). The width of this cline, i.e. the geographic distance over which the allele frequency changes, depends on the relative strengths of dispersal and selection. If selection is strong compared to dispersal, then selection acts to remove maladaptive alleles much faster than migration acts to move alleles across the environmental transition. Thus the allele frequency transition would be very rapid, and the cline narrow, as we move across the environmental transition. In contrast, if individuals disperse long distances and selection is weak, many alleles are being moved back and forth over the environmental transition much faster than selection can act against these alleles and so the cline would be very wide.
The width of our cline, i.e. the distance over which we make this shift from allele \(2\) to allele \(1\) predominating, can be defined in a number of different ways. One way to define the cline width, which is simple to define but perhaps hard to measure accurately, is via the slope (i.e. the tangent) of \(q(x)\) at \(x=0\) . See Figure \(\PageIndex{12}\). Under this definition, the cline width is approximately
\[0.6 \sigma/\sqrt{s} ~\textrm{miles}, \label{eqn:cline_width}\]
note that the units are miles here just because we defined the average dispersal distance ( \(\sigma\) ) in miles above. Thus the cline will be wider if individuals dispersal further, higher \(\sigma\) , and if selection is weaker, smaller \(s\) . The appendix at the end of this chapter, talks through the math underlying these ideas in more detail.
collected mosquitoes ( Culex pipiens ) in a north–south transect moving away from the Southern French coast. Areas near the coast were are treated with pesticides, and the mosquitos have evolved resistance, but areas just a few tens of kilometers from the coast were untreated. estimated the frequency of two unlinked, pesticide-resistance alleles, and found them at high frequency near the coast but found that their frequencies declined rapidly moving inland. fit migration-selection cline models to their data, similar to those in Figure \(\PageIndex{11}\), with the pesticide-resistance alleles having an selection advantage ( \(s\) ) in treated areas an a cost ( \(c\) ) in untreated areas (they didn’t force the selective advantage and cost to be symmetric).
They estimated that a higher selective advantage for the Ace 1 allele than Ester allele ( \(s=0.33\) and \(s=0.19\) respectively) and a higher cost to the Ace 1 allele than Ester allele in untreated areas ( \(c=0.11\) and \(c=07\) respectively) potentially explaining the less extreme cline for Ester allele than the Ace 1 allele. Despite these strong selection pressures, we still see a cline over tens of kilometers because dispersal is relatively high ( \(\sigma= 6.6\) km per generation).
Hybrid zones
Local adaptation isn’t the only way that selection can generate strong spatial patterns. We can also see strong selection-driven clines when partially-reproductively isolated species spread back in to secondary contact they can hybridize bringing alleles together that may not work well with each other. One simple model of is to think about an under-dominant polymorphism, i.e. where the heterozygote has lower fitness. The two ancestral populations are alternatively fixed for the two fitter homozygote states, e.g. ancestral population 1 fixed \(A_1A_1\) and ancestral population two the \(A_2 A_2\) . The hybrid population forming at the mating edge between the two ancestral populations has a high frequency of the less fit heterozygotes. Thus hybrids are at a disadvantage, potentially acting to keep the two populations from collapsing into each other.
Two previously isolated populations of the short-horned grasshopper Podisma pedestris have spread into secondary contact in the French Alps, probably after the last ice age. The population that has spread into the Alps from the south has a large section of novel X chromosome, due to a chromosomal fusion. This ‘neo-X’ is absent in the populations that spread from the North into the Alps. The two populations meet in many valleys running through the Alps, and repeatedly form a narrow hybrid zone, with the frequency of the neo-X chromosome forming a very steep cline transitioning in frequency over a few hundred meters . One potential reason for this steep cline is that females who are heterozygous for the neo-X (neo-X/old-X) may have reduced fitness, consistent with an underdominant polymorphism. The neo-X allele cannot spread into the northern population as it cannot increase in frequency when rare. Conversely the northern population cannot displace the neo-X, as the old-X is at a disadvantage. This spatial distribution at this locus is a tension zone between the two populations, where neither allele can push the other out due to the low fitness of the hybrid.
We can use our same continuous model of migration and selection to study this setup. Assuming that the homozygotes are equally fit, and that the heterozygotes relative fitness is reduced by a selection coefficent \(s_h\) , the width of the cline is
\[\frac{\sigma}{\sqrt{s_h}}\]
The stronger the selection the more abrupt the transition between the populations. These wingless grasshoppers move \(\sigma \sim 20\) meters a generation. Thus a reduction in the relative fitness of the hybrid would be needed to explain this hybrid zone with a width of \(\sim 800\) m.
More generally we can see tension zones arise when hybrids have reduced fitness compared to either species. For example, this can occur due to be due to bad epistatic interactions between alleles from each species. If selection is strong enough on hybrids, often because many loci are involved in incompatibilities between the species, the entire genome can be tied up in a tension zone between the two species.
Appendix: Some theory of the spatial distribution of allele frequencies under deterministic models of selection
Imagine a continuous haploid population spread out along a line. Each individual disperses a random displacement \(\Delta x\) from its birthplace to the location where it reproduces, where \(\Delta x\) is drawn from the probability density \(g(\Delta x)\) . To make life simple, we will assume that \(g(\Delta x)\) is normally distributed with mean zero and standard deviation \(\sigma\) , i.e. migration is unbiased and individuals migrate an average displacement of \(\sigma\) .
The frequency of allele \(2\) at time \(t\) in the population at spatial location \(x\) is \(q(x,t)\) . Assuming that only dispersal occurs, how does our allele frequency change in the next generation? Our allele frequency in the next generation at location \(x\) reflects the migration from different locations in the proceeding generation. Our population at location \(x\) receives a contribution \(g(\Delta x)q(x+\Delta x,t)\) of allele \(2\) from the population at location \(x+\Delta x\) , such that the frequency of our allele at \(x\) in the next generation is
\[q(x,t+1) = \int_{-\infty}^{\infty} g(\Delta x)q(x+\Delta x,t) d \Delta x.\]
To obtain \(q(x+\Delta x,t)\) , let’s take a Taylor series expansion of \(q(x, t)\) :
\[q(x+\Delta x,t) = q(x,t) + \Delta x \frac{dq(x,t)}{dx}+ \tfrac{1}{2}(\Delta x)^2 \frac{d^2q(x,t)}{dx^2}+\cdots\]
then
\[q(x,t+1) = q(x,t) +\left( \int_{-\infty}^{\infty} \Delta x g(\Delta x) d \Delta x \right) \frac{dq(x,t)}{dx} + \tfrac{1}{2}\left( \int_{-\infty}^{\infty}(\Delta x)^2 g(\Delta x) d \Delta x \right) \frac{d^2q(x,t)}{dx^2}+\cdots\]
Because \(g(~)\) has a mean of zero, \(\int_{-\infty}^{\infty} \Delta x g(\Delta x) d \Delta x =0\) , and has because \(g(~)\) has variance \(\sigma^2\) , \(\int_{-\infty}^{\infty}(\Delta x)^2 g(\Delta x) d \Delta x = \sigma^2\) . All higher order terms in our Taylor series expansion cancel out (as all higher central moments of the normal distribution are zero). Looking at the change in allele frequency, \(\Delta q(x,t) = q(x,t+1)-q(x,t)\) , so
\[\Delta q(x,t) = \frac{\sigma^2}{2} \frac{d^2q(x,t)}{dx^2}\]
This is a diffusion equation, so that migration is acting to smooth out allele frequency differences with a diffusion constant of \(\tfrac{\sigma^2}{2}\) . This is exactly analogous to the equation describing how a gas diffuses out to equal density, as both particles in a gas and our individuals of type \(2\) are performing Brownian motion (blurring our eyes and seeing time as continuous).
We will now introduce fitness differences into our model and set the relative fitnesses of allele \(1\) and \(2\) at location \(x\) to be \(1\) and \(1+s\gamma(x)\) . To make progress in this model, we’ll have to assume that selection isn’t too strong, i.e. \(s \gamma(x) \ll 1\) for all \(x\) . The change in frequency of allele \(2\) obtained within a generation due to selection is
\[q^{\prime}(x,t) - q(x,t) \approx s\gamma(x) q(x,t) \big( 1 - q(x,t) \big)\]
i.e. logistic growth of our favoured allele at location \(x\) . Putting our selection and migration terms together, we find the total change in allele frequency at location x in one generation is
\[q(x,t+1) - q(x,t) = s\gamma(x) q(x,t) \big( 1 - q(x,t) \big)+\frac{\sigma^2}{2} \frac{d^2q(x,t)}{dx^2} \label{eqn:fisherKPP}\]
In deriving this result we have ignored terms of the order of \(\sigma s\) .
The cline in allele frequency associated with a sharp environmental transition.
To make progress, let’s consider a simple model of local adaptation where the environment abruptly changes. Specifically, we assume that \(\gamma(x)= 1\) for \(x<0\) and \(\gamma(x)= -1\) for \(x \geq 0\) , i.e. our allele \(2\) has a selective advantage at locations to the left of zero, while this allele is at a disadvantage to the right of zero. In this case we can get an equilibrium distribution of our two alleles, where to the left of zero our allele \(2\) is at higher frequency, while to the right of zero allele \(1\) predominates. As we cross from the left to the right side of our range, the frequency of our allele \(2\) decreases in a smooth cline.
Our equilibrium spatial distribution of allele frequencies can be found by setting the left-hand side of Equation [eqn:fisherKPP] to zero to arrive at
\[s\gamma(x) q(x) \left( 1 - q(x) \right) = - \frac{\sigma^2}{2} \frac{d^2q(x)}{dx^2}\]
We then could solve this differential equation with appropriate boundary conditions ( \(q(-\infty)=1\) and \(q(\infty) = 0\) ) to arrive at the appropriate functional form for our cline. While we won’t go into the solution of this equation here, we can note that by dividing our distance \(x\) by \(\ell=\sigma/\sqrt{s}\) , we can remove the effect of our parameters from the above equation. This compound parameter \(\ell\) is the characteristic length of our cline, and it is this parameter which determines over what geographic scale we change from allele \(2\) predominating to allele \(1\) predominating as we move across our environmental shift.
Summery
- Deleterious variation can be maintained in the population by a balance of selection and mutation. If the mutations are not completely recessive, the equilibrium frequency of deleterious alleles is given by the ratio of mutation to the selection coefficient against heterozygotes ( \(q_{eq}= \frac{\mu}{hs}\) ). The more recessive an allele the higher frequency it segregates under mutation-selection balance, all else being equal, as they better avoid selection in the heterozygote state.
- While the equilibrium frequency of alleles under mutation-selection balance at any one locus is low, there are many such loci in the genome such that every individual carries many deleterious alleles.
- As more recessive deleterious alleles segregate at higher frequency, inbred individuals are expected to have lower fitness than typical outbred individuals in the population as they are on average homozygous for recessive deleterious alleles.
- Divergent selection between populations can maintain allele frequency differences between populations in the face of migration. The constant influx of alleles by migration can maintain maladaptive alleles at low frequency in the face of selection leading to a migration selection balance, an analog to mutation selection balance.
- When strong selection pressures change over short geographical scales, we expect abrupt allele frequency clines at the selected loci. We also expect strong allele frequency clines in hybrid zones at loci underpinning hybrid fitness disadvantage.
You are studying a gene causing partial infertility, due to errors during meiosis, in an outcrossing plant population. You estimate that \(5\%\) of heterozygotes for knockout mutation in this gene are completely sterile, but \(95\%\) of heterozygote individuals have normal fertility. Homozygotes for the knockout are often embryonic lethal due to errors in mitosis. The frequency at birth of knockouts for the gene is \(\frac{1}{5000}\) .
A) What is the knockout mutation rate at this gene?
B) You find a sister species which has had a high degree of inbreeding for many generations due to selfing. Do you expect to find the knockout allele at higher or lower frequency? Explain your answer.
There’s an outbred population of mice living in a farmer’s field. Mutations occur at a gene called nurseryrhyme that cause a totally recessive form of blindness. These blind mice do not survive to reproduce as the farmer’s wife cuts off their tail (and other bits) with a carving knife. Surveying the field for baby mice you find that 3 in ten thousand mice are blind.
A Assuming that the population mates at random, what is the mutation rate of blindness causing alleles?
B Following more careful study you now find that there is actually a \(20 \%\) reduction in the viability of heterozygotes for these mutations. What would you now estimate as the mutation rate for this gene? C) Explain how and why your answers differ?


29.1: Molecular Orbitals of Conjugated Pi Systems
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- Steven Farmer, Dietmar Kennepohl, William Reusch, & Kevin M. Shea
- University of Illinois Springfield
Frontier Molecular Orbital Theory
Prior to 1965, pericyclic reactions were known as "no mechanism reactions" since no one could adequately explain why reaction outcomes changed depending on whether reactants were exposed to heat or light. In 1965 Robert Burns Woodward and Roald Hoffmann used Frontier Molecular Orbital Theory , initially proposed by Kenichi Fukui, to develop their Theory of Conservation of Orbital Symmetry where outcomes of pericyclic reactions are explained by examining the Highest Occupied Molecular Orbital (HOMO) or Lowest Unoccupied Molecular Orbital (LUMO) of the reacting system. Their analysis of cycloadditions, electrocyclic reactions, and sigmatropic rearrangements is commonly referred to as the Woodward-Hoffmann Rules. A detailed analysis of three reaction types is provided in the subsequent sections of this chapter.
HOMO and LUMO are often referred to as frontier orbitals and their energy difference is termed the HOMO–LUMO gap . One common way of thinking about reactions in this way is through the concept of frontier orbitals. This idea says that if one species is going to donate electrons to another in order to form a new bond, then the donated electrons are most likely going to come from the highest occupied energy level. In this level, called the highest occupied molecular orbital (HOMO), the electrons are further from the nucleus and therefore less tightly held by the protons in the nucleus. The electrons would be donated, in turn, to the lowest empty energy level on the other species, called the lowest unoccupied molecular orbital (LUMO).
Molecular orbital interaction between frontier orbitals.
Molecular Orbitals
According to MO theory discussed in Section 1-11 , when a double bond is non-conjugated, the two atomic 2 p z orbitals combine to form two pi ( π ) molecular orbitals , one a low-energy π bonding orbital and one a high-energy π -star ( π *) anti-bonding molecular orbital . These are sometimes denoted, in MO diagrams like the one below, with the Greek letter psi ( Ψ ) instead of π . In the bonding Ψ 1 orbital, the two (+) lobes of the 2p z orbitals interact constructively with each other, as do the two (-) lobes. Therefore, there is increased electron density between the nuclei in the molecular orbital – this is why it is a bonding orbital. In the higher-energy anti-bonding Ψ 2 * orbital, the (+) lobes of one 2 p z orbital interacts destructively with the (-) lobe of the second 2 p z orbital, leading to a node between the two nuclei and overall repulsion. By the aufbau principle, the two electrons from the two atomic orbitals will be paired in the lower-energy Ψ 1 orbital when the molecule is in the ground state.
With a conjugated diene, such as 1,3-butadiene, the four 2p atomic orbitals combine to form four pi molecular orbitals of increasing energy. Two bonding pi orbitals and two antibonding pi* orbitals. The combination of four pi molecular orbitals allow for the formation of a bonding molecular orbital that is lower in energy than those created by an unconjugated alkene. The 4 pi electrons of 1,3-butadiene completely fill the bonding molecular orbitals giving is the additional stability associated with conjugated double bonds.
Electronic Transitions
When a double-bonded molecule such as ethene (common name ethylene) absorbs 165 nm light, it undergoes a π - π * transition . An electron is moved from the HOMO of ethene to the LUMO placing the molecule in an excited state .
Where electronic transition becomes useful to most organic and biological chemists is in the study of molecules with conjugated pi systems. In these groups, the HOMO–LUMO gap energy gap for π - π * transitions is smaller than for isolated double bonds, and thus the wavelength absorbed is longer. The MO diagram for 1,3-butadiene, the simplest conjugated system. Recall that we can draw a diagram showing the four pi MO’s that result from combining the four 2p z atomic orbitals. The lower two orbitals are pi bonding, while the upper two are pi antibonding. Comparing this MO picture to that of ethene, our isolated pi-bond example the HOMO would be psi 2 and the LUMO would be psi 3. The HOMO-LUMO energy gap is smaller for the conjugated 1,3-butadiene system which absorbs UV light with a wavelength of 217 nm.
As conjugated pi systems become larger, the HOMO–LUMO gap energy gap for a π - π * transition becomes increasingly narrow, and the wavelength of light absorbed correspondingly becomes longer. The absorbance due to the π - π * transition in 1,3,5-hexatriene, for example, occurs at 258 nm.
Woodward-Hoffmann Rules
Much of what we have said about the electronic factors controlling pericyclic reaction was formulated in the mid 1960's by the American chemists R. B. Woodward and R. Hoffmann, in terms of what came to be called the orbital symmetry principles, or the Woodward-Hoffmann rules . This is a particularly simple approach says that many details of pericyclic reactions can explained by "conservation of orbital symmetry." This requires the symmetries of the molecular orbitals of reactants to be the same as the molecular orbitals of the products for a reaction to proceed.
The original approach of Woodward and Hoffmann involved construction of an "orbital correlation diagram" to see if the lobes of the reactant molecular orbitals match phases and allow for overlap required for bonding to occur. The symmetries of the appropriate reactant and product orbitals were matched to determine whether the transformation could proceed without a symmetry imposed conversion of bonding reactant orbitals to antibonding product orbitals. If the correlation diagram indicated that the reaction could occur without encountering such a symmetry-imposed barrier , it was termed symmetry allowed . If a symmetry barrier was present, the reaction was designated symmetry-forbidden .
Exercise \(\PageIndex{1}\)
Using the molecular orbital diagram for 1,3,5-hexatriene determine the HOMO and LUMO for both the ground and excited state.
- Answer
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For the ground state the HOMO is psi 3 and LUMO is psi 4.
For the excited state the HOMO is psi 4 and LUMO is psi 5.


8.11: References
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10.4: Trigonometric Functions of Any Angle
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To define the trigonometric functions of any angle - including angles less than \(0^\circ\) or greater than \(360^\circ \) - we need a more general definition of an angle. We say that an angle is formed by rotating a ray \(\overrightarrow{OA} \) about the endpoint \(O \) (called the vertex ), so that the ray is in a new position, denoted by the ray \(\overrightarrow{OB} \). The ray \(\overrightarrow{OA} \) is called the \(\textbf{initial side}\) of the angle, and \(\overrightarrow{OB} \) is the terminal side of the angle (see Figure 1.4.1(a)).
We denote the angle formed by this rotation as \(\angle\,AOB \), or simply \(\angle\,O \), or even just \(O \). If the rotation is counter-clockwise then we say that the angle is positive , and the angle is negative if the rotation is clockwise (see Figure 1.4.1(b)).
One full counter-clockwise rotation of \(\overrightarrow{OA} \) back onto itself (called a revolution ), so that the terminal side coincides with the initial side, is an angle of \(360^\circ\); in the clockwise direction this would be \(-360^\circ \). Not rotating \(\overrightarrow{OA} \) constitutes an angle of \(0^\circ \). More than one full rotation creates an angle greater than \(360^\circ \). For example, notice that \(30^\circ \) and \(390^\circ \) have the same terminal side in Figure 1.4.2, since \(30 + 360 = 390 \).
We can now define the trigonometric functions of any angle in terms of \(\textbf{Cartesian coordinates}\). Recall that the \(\mathbf{xy}\) -coordinate plane consists of points denoted by pairs \((x,y) \) of real numbers. The first number, \(x \), is the point's \(\mathbf{x} \) coordinate , and the second number, \(y \), is its \(\mathbf{y} \) coordinate . The \(x \) and \(y \) coordinates are measured by their positions along the \(\mathbf{x}\)- axis and \(\mathbf{y}\)- axis , respectively, which determine the point's position in the plane. This divides the \(xy\)-coordinate plane into four quadrants (denoted by QI, QII, QIII, QIV), based on the signs of \(x \) and \(y \) (see Figure 1.4.3(a)-(b)).
Now let \(\theta \) be any angle. We say that \(\theta \) is in standard position if its initial side is the positive \(x\)-axis and its vertex is the origin \((0,0) \). Pick any point \((x,y) \) on the terminal side of \(\theta \) a distance \(r>0 \) from the origin (see Figure 1.4.3(c)). (Note that \(r = \sqrt{ x^2 + y^2 } \). Why?) We then define the trigonometric functions of \(\theta \) as follows:
\[ \sin\;\theta ~=~ \dfrac{y}{r} \qquad\qquad
\cos\;\theta ~=~ \dfrac{x}{r} \qquad\qquad
\tan\;\theta ~=~ \dfrac{y}{x} \label{1.2}\]
\[\csc\;\theta ~=~ \dfrac{r}{y} \qquad\qquad
\sec\;\theta ~=~ \dfrac{r}{x} \qquad\qquad
\cot\;\theta ~=~ \dfrac{x}{y} \label{1.3}\]
As in the acute case, by the use of similar triangles these definitions are well-defined (i.e. they do not depend on which point \((x,y) \) we choose on the terminal side of \(\theta\)). Also, notice that \(| \sin\;\theta | \le 1 \) and \( | \cos\;\theta | \le 1 \), since \(| y | \le r \) and \( | x | \le r \) in the above definitions.
Notice that in the case of an acute angle these definitions are equivalent to our earlier definitions in terms of right triangles: draw a right triangle with angle \(\theta \) such that \(x = \text{adjacent side} \), \(y = \text{opposite side} \), and \(r = \text{hypotenuse} \). For example, this would give us \(\sin\;\theta = \frac{y}{r} = \frac{\text{opposite}}{\text{hypotenuse}} \) and \(\cos\;\theta = \frac{x}{r} = \frac{\text{adjacent}}{\text{hypotenuse}} \), just as before (see Figure 1.4.4(a)).
In Figure 1.4.4(b) we see in which quadrants or on which axes the terminal side of an angle \(0^\circ \le \theta < 360^\circ \) may fall. From Figure 1.4.3(a) and formulas Equation \ref{1.2} and Equation \ref{1.3}, we see that we can get negative values for a trigonometric function. For example, \(\sin\;\theta < 0 \) when \(y<0 \). Figure 1.4.5 summarizes the signs (positive or negative) for the trigonometric functions based on the angle's quadrant:
Example 1.20
Find the exact values of all six trigonometric functions of \(120^\circ\).
Solution
We know \(120^\circ = 180^\circ - 60^\circ\). By Example 1.7 in Section 1.2, we see that we can use the point \((-1,\sqrt{3})\) on the terminal side of the angle \(120^\circ\) in QII, since we saw in that example that a basic right triangle with a \(60^\circ\) angle has adjacent side of length \(1\), opposite side of length \(\sqrt{3}\), and hypotenuse of length \(2\), as in the figure on the right. Drawing that triangle in QII so that the hypotenuse is on the terminal side of \(120^\circ\) makes \(r = 2\), \(x=-1\), and \(y=\sqrt{3}\). Hence:
\[\nonumber \sin\;120^\circ \;=\; \dfrac{y}{r} \;=\; \dfrac{\sqrt{3}}{2} \qquad
\cos\;120^\circ \;=\; \dfrac{x}{r} \;=\; \dfrac{-1}{2} \qquad
\tan\;120^\circ \;=\; \dfrac{y}{x} \;=\; \dfrac{\sqrt{3}}{-1} \,=\, -\sqrt{3} \nonumber \]
\[\nonumber \csc\;120^\circ \;=\; \dfrac{r}{y} \;=\; \dfrac{2}{\sqrt{3}} \qquad
\sec\;120^\circ \;=\; \dfrac{r}{x} \;=\; \dfrac{2}{-1} \;=\; -2 \qquad
\cot\;120^\circ \;=\; \dfrac{x}{y} \;=\; \dfrac{-1}{\sqrt{3}} \nonumber \]
Example 1.21
Find the exact values of all six trigonometric functions of \(225^\circ \).
Solution
We know that \(225^\circ = 180^\circ + 45^\circ \). By Example 1.6 in Section 1.2, we see that we can use the point \((-1,-1) \) on the terminal side of the angle \(225^\circ \) in QIII, since we saw in that example that a basic right triangle with a \(45^\circ \) angle has adjacent side of length \(1 \), opposite side of length \(1 \), and hypotenuse of length \(\sqrt{2} \), as in the figure on the right. Drawing that triangle in QIII so that the hypotenuse is on the terminal side of \(225^\circ \) makes \(r = \sqrt{2} \), \(x=-1 \), and \(y=-1 \). Hence:
\[\nonumber \sin\;225^\circ \;=\; \dfrac{y}{r} \;=\; \dfrac{-1}{\sqrt{2}} \qquad
\cos\;225^\circ \;=\; \dfrac{x}{r} \;=\; \dfrac{-1}{\sqrt{2}} \qquad
\tan\;225^\circ \;=\; \dfrac{y}{x} \;=\; \dfrac{-1}{-1} \,=\, 1 \nonumber \]
\[\nonumber \csc\;225^\circ \;=\; \dfrac{r}{y} \;=\; -\sqrt{2} \qquad
\sec\;225^\circ \;=\; \dfrac{r}{x} \;=\; -\sqrt{2} \qquad
\cot\;225^\circ \;=\; \dfrac{x}{y} \;=\; \dfrac{-1}{-1} \,=\, 1 \nonumber \]
Example 1.22
Find the exact values of all six trigonometric functions of \(330^\circ \).
Solution
We know that \(330^\circ = 360^\circ - 30^\circ \). By Example 1.7 in Section 1.2, we see that we can use the point \((\sqrt{3},-1) \) on the terminal side of the angle \(225^\circ \) in QIV, since we saw in that example that a basic right triangle with a \(30^\circ \) angle has adjacent side of length \(\sqrt{3} \), opposite side of length \(1 \), and hypotenuse of length \(2 \), as in the figure on the right. Drawing that triangle in QIV so that the hypotenuse is on the terminal side of \(330^\circ \) makes \(r = 2 \), \(x=\sqrt{3} \), and \(y=-1 \). Hence:
\[\nonumber \sin\;330^\circ \;=\; \dfrac{y}{r} \;=\; \dfrac{-1}{2} \qquad
\cos\;330^\circ \;=\; \dfrac{x}{r} \;=\; \dfrac{\sqrt{3}}{2} \qquad
\tan\;330^\circ \;=\; \dfrac{y}{x} \;=\; \dfrac{-1}{\sqrt{3}} \nonumber \]
\[\nonumber \csc\;330^\circ \;=\; \dfrac{r}{y} \;=\; -2 \qquad
\sec\;330^\circ \;=\; \dfrac{r}{x} \;=\; \dfrac{2}{\sqrt{3}} \qquad
\cot\;330^\circ \;=\; \dfrac{x}{y} \;=\; -\sqrt{3} \nonumber \]
Example 1.23
Find the exact values of all six trigonometric functions of \(0^\circ \), \(90^\circ \), \(180^\circ \), and \(270^\circ \).
Solution
These angles are different from the angles we have considered so far, in that the terminal sides lie along either the \(x\)-axis or the \(y\)-axis. So unlike the previous examples, we do not have any right triangles to draw. However, the values of the trigonometric functions are easy to calculate by picking the simplest points on their terminal sides and then using the definitions in formulas Equation \ref{1.2} and Equation \ref{1.3}.
For instance, for the angle \(0^\circ \) use the point \((1,0) \) on its terminal side (the positive \(x\)-axis), as in Figure 1.4.6. You could think of the line segment from the origin to the point \((1,0) \) as sort of a degenerate right triangle whose height is \(0 \) and whose hypotenuse and base have the same length \(1 \). Regardless, in the formulas we would use \(r = 1 \), \(x = 1 \), and \(y = 0 \). Hence:
\[\nonumber \sin\;0^\circ \;=\; \dfrac{y}{r} \;=\; \dfrac{0}{1} \;=\; 0 \qquad
\cos\;0^\circ \;=\; \dfrac{x}{r} \;=\; \dfrac{1}{1} \;=\; 1 \qquad
\tan\;0^\circ \;=\; \dfrac{y}{x} \;=\; \dfrac{0}{1} \;=\; 0 \nonumber \]
\[\nonumber \csc\;0^\circ \;=\; \dfrac{r}{y} \;=\; \dfrac{1}{0} \;=\; \text{undefined}\qquad
\sec\;0^\circ \;=\; \dfrac{r}{x} \;=\; \dfrac{1}{1} \;=\; 1 \qquad
\cot\;0^\circ \;=\; \dfrac{x}{y} \;=\; \dfrac{1}{0} \;=\; \text{undefined} \nonumber \]
Note that \(\csc\;0^\circ \) and \(\cot\;0^\circ \) are undefined, since division by \(0 \) is not allowed.
Similarly, from Figure 1.4.6 we see that for \(90^\circ \) the terminal side is the positive \(y\)-axis, so use the point \((0,1) \). Again, you could think of the line segment from the origin to \((0,1) \) as a degenerate right triangle whose base has length \(0 \) and whose height equals the length of the hypotenuse. We have \(r = 1 \), \(x = 0 \), and \(y = 1 \), and hence:
\[\nonumber \sin\;90^\circ \;=\; \dfrac{y}{r} \;=\; \dfrac{1}{1} \;=\; 1 \qquad
\cos\;90^\circ \;=\; \dfrac{x}{r} \;=\; \dfrac{0}{1} \;=\; 0 \qquad
\tan\;90^\circ \;=\; \dfrac{y}{x} \;=\; \dfrac{1}{0} \;=\; \text{undefined} \nonumber \]
\[\nonumber \csc\;90^\circ \;=\; \dfrac{r}{y} \;=\; \dfrac{1}{1} \;=\; 1\qquad
\sec\;90^\circ \;=\; \dfrac{r}{x} \;=\; \dfrac{1}{0} \;=\; \text{undefined}\qquad
\cot\;90^\circ \;=\; \dfrac{x}{y} \;=\; \dfrac{0}{1} \;=\; 0 \nonumber \]
Likewise, for \(180^\circ \) use the point \((-1,0) \) so that \(r = 1 \), \(x = -1 \), and \(y = 0 \). Hence:
\[\nonumber \sin\;180^\circ \;=\; \dfrac{y}{r} \;=\; \dfrac{0}{1} \;=\; 0 \qquad
\cos\;180^\circ \;=\; \dfrac{x}{r} \;=\; \dfrac{-1}{1} \;=\; -1 \qquad
\tan\;180^\circ \;=\; \dfrac{y}{x} \;=\; \dfrac{0}{-1} \;=\; 0 \nonumber \]
\[\nonumber \csc\;180^\circ \;=\; \dfrac{r}{y} \;=\; \dfrac{1}{0} \;=\; \text{undefined}\quad\;\;\;
\sec\;180^\circ \;=\; \dfrac{r}{x} \;=\; \dfrac{1}{-1} \;=\; -1\quad\;\;\;
\cot\;180^\circ \;=\; \dfrac{x}{y} \;=\; \dfrac{-1}{0} \;=\; \text{undefined} \nonumber \]
Lastly, for \(270^\circ \) use the point \((0,-1) \) so that \(r = 1 \), \(x = 0 \), and \(y = -1 \). Hence:
\[\nonumber \sin\;270^\circ \;=\; \dfrac{y}{r} \;=\; \dfrac{-1}{1} \;=\; -1 \qquad
\cos\;270^\circ \;=\; \dfrac{x}{r} \;=\; \dfrac{0}{1} \;=\; 0 \qquad
\tan\;270^\circ \;=\; \dfrac{y}{x} \;=\; \dfrac{-1}{0} \;=\; \text{undefined} \nonumber \]
\[\nonumber \csc\;270^\circ \;=\; \dfrac{r}{y} \;=\; \dfrac{1}{-1} \;=\; -1\qquad
\sec\;270^\circ \;=\; \dfrac{r}{x} \;=\; \dfrac{1}{0} \;=\; \text{undefined}\qquad
\cot\;270^\circ \;=\; \dfrac{x}{y} \;=\; \dfrac{0}{-1} \;=\; 0 \nonumber \]
The following table summarizes the values of the trigonometric functions of angles between \(0^\circ \) and \(360^\circ \) which are integer multiples of \(30^\circ \) or \(45^\circ\):
Table 1.3 Table of trigonometric function values
Since \(360^\circ \) represents one full revolution, the trigonometric function values repeat every \(360^\circ \). For example, \(\sin\;360^\circ = \sin\;0^\circ \), \(\cos\;390^\circ = \cos\;30^\circ \), \(\tan\;540^\circ = \tan\;180^\circ \), \(\sin\;(-45^\circ) = \sin\;315^\circ \), etc. In general, if two angles differ by an integer multiple of \(360^\circ\) then each trigonometric function will have equal values at both angles. Angles such as these, which have the same initial and terminal sides, are called coterminal .
In Examples 1.20-1.22, we saw how the values of trigonometric functions of an angle \(\theta \) larger than \(90^\circ \) were found by using a certain acute angle as part of a right triangle. That acute angle has a special name: if \(\theta \) is a nonacute angle then we say that the reference angle for \(\theta \) is the acute angle formed by the terminal side of \(\theta \) and either the positive or negative \(x\)-axis. So in Example 1.20, we see that \(60^\circ \) is the reference angle for the nonacute angle \(\theta = 120^\circ\); in Example 1.21, \(45^\circ \) is the reference angle for \(\theta = 225^\circ\); and in Example 1.22, \(30^\circ \) is the reference angle for \(\theta = 330^\circ \).
Example 1.24
- Which angle between \(0^\circ \) and \(360^\circ \) has the same terminal side (and hence the same trigonometric function values) as \(\theta\,\)?
- What is the reference angle for \(\theta\,\)?
Solution
(a) Since \(928^\circ = 2 \times 360^\circ + 208^\circ \), then \(\theta \) has the same terminal side as \(208^\circ \), as in Figure 1.4.7.
(b) \(928^\circ \) and \(208^\circ \) have the same terminal side in QIII, so the reference angle for \(\theta = 928^\circ \) is \(208^\circ - 180^\circ = 28^\circ \).
Example 1.25
Suppose that \(\cos\;\theta = -\frac{4}{5} \). Find the exact values of \(\sin\;\theta \) and \(\tan\;\theta \).
Solution
We can use a method similar to the one used to solve Example 1.8 in Section 1.2. That is, draw a right triangle and interpret \(\cos\;\theta \) as the ratio \(\frac{\text{adjacent}}{\text{hypotenuse}} \) of two of its sides. Since \(\cos\;\theta = -\frac{4}{5} \), we can use \(4 \) as the length of the adjacent side and \(5 \) as the length of the hypotenuse. By the Pythagorean Theorem, the length of the opposite side must then be \(3 \). Since \(\cos\;\theta \) is negative, we know from Figure 1.4.5 that \(\theta \) must be in either QII or QIII. Thus, we have two possibilities, as shown in Figure 1.4.8 below:
When \(\theta \) is in QII, we see from Figure 1.4.8(a) that the point \((-4,3) \) is on the terminal side of \(\theta \), and so we have \(x = -4 \), \(y = 3 \), and \(r = 5 \). Thus, \(\sin\;\theta = \frac{y}{r} = \frac{3}{5} \) and \(\tan\;\theta = \frac{y}{x} = \frac{3}{-4} \).
When \(\theta \) is in QIII, we see from Figure 1.4.8(b) that the point \((-4,-3) \) is on the terminal side of \(\theta \), and so we have \(x = -4 \), \(y = -3 \), and \(r = 5 \). Thus, \(\sin\;\theta = \frac{y}{r} = \frac{-3}{5} \) and \(\tan\;\theta = \frac{y}{x} = \frac{-3}{-4} =
\frac{3}{4} \).
Thus, either \(\fbox{\(\sin\;\theta = \frac{3}{5} \) and \(\tan\;\theta = -\frac{3}{4}\)}\) or \(\fbox{\(\sin\;\theta = -\frac{3}{5} \) and \(\tan\;\theta = \frac{3}{4}\)}\).
Since reciprocals have the same sign, \(\csc\;\theta \) and \(\sin\;\theta \) have the same sign, \(\sec\;\theta \) and \(\cos\;\theta \) have the same sign, and \(\cot\;\theta \) and \(\tan\;\theta \) have the same sign. So it suffices to remember the signs of \(\sin\;\theta \), \(\cos\;\theta \), and \(\tan\;\theta\):
For an angle \(θ\) in standard position and a point \((x, y)\) on its terminal side:
- \(\sin\;\theta \) has the same sign as \(y\)
- \(\cos\;\theta \) has the same sign as \(x\)
- \(\tan\;\theta \) is positive when \(x \) and \(y \) have the same sign
- \(\tan\;\theta \) is negative when \(x \) and \(y \) have opposite signs
Contributors
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Michael Corral (Schoolcraft College). The content of this page is distributed under the terms of the GNU Free Documentation License, Version 1.2 .


8.2: Conservative Interactions
Let's summarize the physical concepts and principles we have encountered so far in our study of classical mechanics. We have “discovered” one important quantity, the inertia or inertial mass of an object, and introduced two different quantities based on that concept, the momentum \(m \vec v\) and the kinetic energy \(\frac{1}{2}mv^2\). We found that these quantities have different but equally intriguing properties. The total momentum of a system is insensitive to the interactions between the parts that make up the system, and therefore it stays constant in the absence of external influences (a more general statement of the law of inertia, the first important principle we encountered). The total kinetic energy, on the other hand, changes while any sort of interaction is taking place, but in some cases it may actually return to its original value afterwards. This chapter deals with interactions from an energy point of view, whereas later chapters will deal with them from a force point of view.
In the previous chapter I suggested that what was going on in an elastic collision could be interpreted, or described (perhaps in a figurative way) more or less as follows: as the objects come together, the total kinetic energy goes down, but it is as if it was being temporarily stored away somewhere, and as the objects separate, that “stored energy” is fully recovered as kinetic energy. Whether this does happen or not in any particular collision (that is, whether the collision is elastic or not) depends, as we have seen, on the kind of interaction (“bouncy” or “sticky,” for instance) that takes place between the objects.
We are going to take the above description literally, and use the name conservative interaction for any interaction that can “store and restore” kinetic energy in this way. The “stored energy” itself—which is not actually kinetic energy while it remains stored, since it is not given by the value of \(\frac{1}{2}mv^2\) at that time—we are going to call potential energy . Thus, conservative interactions will be those that have a “potential energy” associated with them, and vice-versa.
Potential Energy
Perhaps the simplest and clearest example of the storage and recovery of kinetic energy is what happens when you throw an object straight upwards, as it rises and eventually falls back down. The object leaves your hand with some kinetic energy; as it rises it slows down, so its kinetic energy goes down, down... all the way down to zero, eventually, as it momentarily stops at the top of its rise. Then it comes down, and its kinetic energy starts to increase again, until eventually, as it comes back to your hand, it has very nearly the same kinetic energy it started out with (exactly the same, actually, if you neglect air resistance).
The interaction responsible for this change in the object’s kinetic energy is, of course, the gravitational interaction between it and the Earth, so we are going to say that the “missing” kinetic energy is temporarily stored as gravitational potential energy of the system formed by the Earth and the object. This potential energy takes a simple mathematical form,
\[ U_g=mgy, \]
where \(y\) is the height of the object above the Earth. We have carefully called this height the coordinate \(y\) to emphasize that is can be negative if \(y\) is negative. It turns out that with this definition of potential energy, the total energy in freefall is
\[ K + U^G = \mathrm{constant} \label{eq:5.4} .\]
This is a statement of conservation of energy under the gravitational interaction. For any interaction that has a potential energy associated with it, the quantity \(K + U\) is called the (total) mechanical energy .
Figure \(\PageIndex{1}\) shows how the kinetic and potential energies of an object thrown straight up change with time (don't worry where this plot comes from yet, you will learn about that when we study projectile motion in detail). I have arbitrarily assumed that the object has a mass of 1 kg and an initial velocity of 2 m/s, and it is thrown from an initial height of 0.5 m above the ground. Note how the change in potential energy exactly mirrors the change in kinetic energy (so \(\Delta U^G = −\Delta K\), mathematically), and the total mechanical energy remains equal to its initial value of 6.9 J throughout.
There is something about potential energy that probably needs to be mentioned at this point. Because I have chosen to launch the object from 0.5 m above the ground, and I have chosen to measure \(y\) from the ground, I started out with a potential energy of \(mgy_i\) = 4.9 J. This makes sense, in a way: it tells you that if you simply dropped the object from this height, it would have picked up an amount of kinetic energy equal to 4.9 J by the time it reached the ground. But, actually, where I choose the vertical origin of coordinates is arbitrary. I could start measuring \(y\) from any height I wanted to—for instance, taking the initial height of my hand to correspond to \(y\) = 0. This would shift the blue curve in Figure \(\PageIndex{1}\) down by 4.9 J, but it would not change any of the physics. The only important thing I really want the potential energy for is to calculate the kinetic energy the object will lose or gain as it moves from one height to another , and for that only changes in potential energy matter. I can always add or subtract any (constant) number to or from \(U\), and it will still be true that \(\Delta K = −\Delta U\).
Closed Systems
Today, physics is pretty much founded on the belief that the energy of a closed system (defined as one that does not exchange energy with its surroundings—more on this in a minute) is always conserved : that is, internal processes and interactions will only cause energy to be “converted” from one form into another, but the total, after all the forms of energy available to the system have been carefully accounted for, will not change. This belief is based on countless experiments, on the one hand, and, on the other, on the fact that all the fundamental interactions that we are aware of do conserve a system’s total energy.
Of course, recognizing whether a system is “closed” or not depends on having first a complete catalogue of all the ways in which energy can be stored and exchanged—to make sure that there is, in fact, no exchange of energy going on with the surroundings. Note, incidentally, that a “closed” system is not necessarily the same thing as an “isolated” system: the former relates to the total energy, the latter to the total momentum. A parked car getting hotter in the sun is not a closed system (it is absorbing energy all the time) but, as far as its total momentum is concerned, it is certainly fair to call it “isolated.” Hopefully all these concepts will be further clarified when we introduce the additional auxiliary concepts of force, work, and heat.
For a closed system, we can state the principle of conservation of energy (somewhat symbolically) in the form
\[ K+U+E_{\text {source }}+E_{\text {diss}}=\text {constant } \label{eq:5.8} \]
where \(K\) is the total, macroscopic, kinetic energy; \(U\) the sum of all the applicable potential energies associated with the system’s internal interactions; \(E_{source}\) is any kind of internal energy (such as chemical energy) that is not described by a potential energy function, but can increase the system’s mechanical energy; and \(E_{diss}\) stands for the contents of the “dissipated energy reservoir”—typically thermal energy. As with the potential energy \(U\), the absolute value of \(E_{source}\) and \(E_{diss}\) does not (usually) really matter: all we are interested in is how much they change in the course of the process under consideration.
Figure \(\PageIndex{1}\) above is an example of this kind of “energy accounting” for a ball bouncing on the ground. If the ball is thrown down, the system formed by the ball and the earth initially has both gravitational potential energy, and kinetic energy (diagram (a)). So all the kinetic energy that we have is the kinetic energy of the ball. As the ball falls, gravitational potential energy is being converted into kinetic energy, and the ball speeds up. As it is about to hit the ground (diagram (b)), the potential energy is zero and the kinetic energy is maximum. During the collision with the ground, all the kinetic energy is temporarily converted into other forms of energy, which are essentially elastic energy of deformation (like the energy in a spring) and some thermal energy (diagram (c)). When it bounces back, its kinetic energy will only be a fraction \(e^2\) of what it had before the collision (where e is the coefficient of restitution). This kinetic energy is all converted into gravitational potential energy as the ball reaches the top of its bounce (diagram (d)). Note there is more dissipated energy in diagram (d) than in (c); this is because I have assumed that dissipation of energy takes place both during the compression and the subsequent expansion of the ball.
Note that in this closed system, some energy was transferred to heat, but the total energy stayed the same, and was therefore conserved. The real problem appears if you can't actually keep track of that energy that gets transferred to heat, then the system would appear not to conserve energy. Mostly, we will assume no transfer to heat in our problems, but we should acknowledge this is not a problem with physics, it is rather a problem with our ability to keep track of all the sources of energy in a system. As you move through the field of physics, you will learn how we keep track of more of these sources of energy, and the problems you solve will become better and better approximations to "the real world".


2.5: Roots of Unity and Related Topics
The complex number \(z=e^{j2π/N}\) is illustrated in Figure \(\PageIndex{1}\). It lies on the unit circle at angle \(θ=2π/N\). When this number is raised to the n th power, the result is \(z^n=e^{j2πn/N}\). This number is also illustrated in Figure \(\PageIndex{1}\). When one of the complex numbers \(e^{j2πn/N}\) is raised to the N th power, the result is
\[(e^{j2πn/N})^N=e^{j2πn}=1 \nonumber \]
We say that \(e^{j2πn/N}\) is one of the \(N\) th roots of unity, meaning that \(e^{j2πn/N}\) is one of the values of \(z\) for which
\[z^N−1=0 \nonumber \]
There are \(N\) such roots, namely,
\[e^{j2πn/N},n=0,1,...,N−1 \nonumber \]
As illustrated in the Figure \(\PageIndex{2}\), the 12 th roots of unity are uniformly distributed around the unit circle at angles \(2πn/12\). The sum of all of the \(N\) th roots of unity is zero:
\[S_N=\sum_{n=0}^{N−1}e^{j2πn/N}=0 \nonumber \]
This property, which is obvious from the Figure, is illustrated in Figure, where the partial sums \(S_k=\sum_{n=0}^{k−1} e^{j2πn/N}\) are plotted for \(k=1,2,...,N\).
These partial sums will become important to us in our study of phasors and light diffraction in "Phasors" and in our discussion of filters in "Filtering".
Geometric Sum Formula
It is natural to ask whether there is an analytical expression for the partial sums of roots of unity:
\[S_k=\sum_{n=0}^{k−1}e^{j2πn/N} \nonumber \]
We can imbed this question in the more general question, is there an analytical solution for the “geometric sum”
\[S_k=\sum_{n=0}^{k−1}z^n? \nonumber \]
The answer is yes, and here is how we find it. If \(z=1\), the answer is \(S_k=k\). If \(z≠1\), we can premultiply \(S_k\) by \(z\) and proceed as follows:
\[\begin{align*} zS_k &=\sum^{k−1}_{n=0}z^{n+1}=\sum^k_{m=1}z^m \\[4pt] &=\sum^{k−1}_{m=0}z^m+z^k−1 \\[4pt] &=S_k+z^k−1 \end{align*} \nonumber \]
From this formula we solve for the geometric sum:
\[S_k=\begin{matrix}\frac {1−z^k} {1−z} & z≠1\\k & z=1\end{matrix} \nonumber \]
This basic formula for the geometric sum \(S_k\) is used throughout electromagnetic theory and system theory to solve problems in antenna design and spectrum analysis. Never forget it.
Find formulas for \(S_k=\sum_{n=0}^{k−1}e^{jnθ}\) and for \(S_k=\sum_{n=0}^{k−1}e^{j2π/Nn}\).
Prove \(\sum_{n=0}^{N−1}e^{j2πn/N}=0\).
Find formulas for the magnitude and phase of the partial sum \(S_k=\sum_{n=0}^{k−1}e^{j2πn/N}\).
(MATLAB) Write a MATLAB program to compute and plot the partial sum \(S_k=\sum_{n=0}^{k−1}e^{j2πn/N}\) for \(k=1,2,...,N\). You should observe the last Figure.
Find all roots of the equation \(z^3+z^2+3z−15=0\).
Find \(c\) so that \((1+j)\) is a root of the equation \(z^{17}+2z^{15}−c=0\).


8.8: Special care placement
Special Education in Juvenile Delinquency Cases Individual Disability Education Act's (IDEA) comprehensive system of identification, evaluation, service delivery, and review has special relevance for juvenile justice professionals. The purpose of the special education system, like the juvenile justice system, is to provide individualized services designed to meet the needs of a particular youth. The enhanced behavioral intervention and transition service needs requirements in the 1997 IDEA amendments bring special education goals even closer to those of the juvenile court. Moreover, the careful documentation of service needs and ongoing assessment of progress required by IDEA bring valuable informational resources to juvenile justice professionals. This section presents a brief overview of how special education information may be helpful as cases make their way through juvenile court.
Some of the issues discussed, such as insanity or incompetence, arise only occasionally. Others, such as the impact on disposition of whether a child has a disability, are relevant in every case in which a delinquent youth is eligible for special education services.
Figure 8.6 Law enforcement transport juveniles to detention facilities such as this to be booked. Security is a concern and officers must enter through a sally port. This is a secure area where the first door is locked before the second door can be opened into the facility. Booking photo by Tabitha Raber is used under a CC BY 4.0 license.
Intake and Initial Interviews
The short time frame for juvenile court proceedings leaves little room for missed opportunities. Juvenile justice professionals must be alert from the earliest moment for clues to the youth’s special education status or existing unidentified disabilities. This process, which should become part of the standard operating procedure, includes carefully interviewing the youth and his or her parents, routinely gathering educational records, procuring examinations by educational and mental health experts, investigating educational services at potential placement facilities, and coordinating juvenile court proceedings with the youth’s IEP team.
Under the 1997 IDEA amendments, whenever a school reports a crime allegedly committed by a youth with a disability, school officials must provide copies of the youth’s special education and disciplinary records to the appropriate authorities to whom the school reports the crime, but only to the extent that the Family Educational Rights and Privacy Act (FERPA) permits the transmission. FERPA allows school officials to transmit school records to law enforcement officials only if parents’ consent in writing to the transmission and in certain other narrowly tailored situations (see 34 C.F.R. § 99.30). This requirement should help ensure that, at least in appropriate school-related cases, special education history, assessments, and service information are readily available early in the court process.
Juvenile justice professionals can learn to recognize disabilities by carefully reading the legal definitions of disability. It is important to understand that youth may have a variety of impairments that are not immediately apparent. Numerous checklists and screening instruments are available to help recognize signs of disabilities and to determine eligibility for special education services (National Council of Juvenile and Family Court Judges, 1991).
If circumstances suggest the need for an eligibility evaluation, modification of a previously existing IEP, or some other exercise of the youth’s rights under special education law, juvenile justice professionals should ensure that appropriate action is expeditiously taken. They should request that parents give written consent for the release of records and should submit a written request for information, evaluation, or review to the LEA.
Juvenile justice professionals could start by contacting the LEA to obtain its policies and procedures for providing special education services to youth in the juvenile justice system. Some districts have designated an individual to deal with compliance issues, and that person may be helpful in expediting or forwarding requests to the right person or agency. Most jurisdictions have a number of other groups that can provide advocacy or other assistance in navigating the special education system. Protection and advocacy offices, special education advocacy groups, learning disabilities associations, and other groups providing support or advocacy for particular disabilities may greatly assist juvenile justice professionals.
Determination of Whether Formal Juvenile Proceedings Should Go Forward
Nothing in IDEA prohibits an agency from “reporting a crime committed by a child with a disability to appropriate authorities” or prevents law enforcement and judicial authorities from “exercising their responsibilities with regard to the application of Federal and State law to crimes committed by a child with a disability. These provisions, outlined in the 1997 amendments, were made in response to concerns that IDEA’s procedural protections could be interpreted to preclude juvenile court jurisdiction over school-related crimes committed by youth with disabilities. In the past, at least one court ruled under State law that a school could not initiate a juvenile court prosecution as a means of evading the procedural requirements of IDEA. Other courts found the juvenile court lacked jurisdiction in cases involving noncriminal school related misconduct in which special education procedures had not been followed.
In at least one case decided after the 1997 amendments, the court confirmed that IDEA does not prevent juvenile courts from exercising jurisdiction over students with disabilities, even if the school is attempting to evade its special education responsibilities. Nonetheless, intake officers and prosecutors should scrutinize whether such evasion has occurred in determining whether a particular case belongs in the juvenile justice system and how it should be processed. Courts and hearing officers have stressed that the school’s responsibility to comply with IDEA procedural requirements does not end when a youth with a disability enters the juvenile justice system.
Even if courts have the power to act, that does not mean the power should be exercised in every case. Long before the 1997 IDEA amendments, a number of courts found that the best course was to dismiss the juvenile court case or defer it until special education proceedings stemming from the misbehavior could be completed.
Many juvenile justice professionals have encountered cases in which a youth enters the juvenile justice system for a relatively minor offense and his or her stay escalates into long-term incarceration because of the youth’s inability to succeed in programs developed for low-risk delinquent youth. This may happen either because the disability-related behavior makes it difficult for the youth to understand or comply with program demands or because his or her behavior is misinterpreted as showing a poor attitude, lack of remorse, or disrespect for authority.
If the juvenile court petition involves a youth with an identified or suspected disability, juvenile justice professionals should first consider whether school-based special education proceedings could provide services or other interventions that would obviate the need for juvenile court proceedings. This is particularly true for incidents occurring at school. The 1997 IDEA amendments require thorough scrutiny of behavioral needs and implementation of appropriate interventions that may far exceed what most juvenile courts are able to provide. In appropriate cases, the juvenile court may wish to consider:
- Continuing or deferring the formal prosecution pending the outcome of special education due process and disciplinary proceedings that may alleviate the need for juvenile court intervention.
- Placing first-time offenders and/or youth alleged to have committed offenses that are not considered too serious for informal handling into diversion or informal supervision programs. Through such programs, the court imposes specific conditions on the youth’s behavior, such as regular school attendance, participation in counseling, observation of specified curfews, or involvement in community service programs. If the youth successfully complies with these conditions, the case is dismissed at the end of a specified period—usually 6 months to 1 year. Allowing the youth to remain in the community, subject to such conditions, may facilitate the completion of special education proceedings while ensuring heightened supervision of the youth. Through IEP development or modification, the youth might be determined eligible for services that supplant the need for formal juvenile court proceedings.
- Dismissing the case in the interest of justice. This option should be considered in cases in which the disability is so severe that it may be difficult or impossible for the youth to comply with court orders. This may occur, for example, if the offense is relatively minor; the youth suffers from mental illness, emotional disturbance, or mental retardation; and/ or services are forthcoming through the special education system.
Think about it . . . Mental health treatment
Many young people in South Carolina have better access to mental health care in juvenile jail than they do outside the fence, back in their own communities. Read these personal interviews of youth in the system in treatment centers. Do you think this approach to treating juvenile delinquents with mental illness instead of jailing them is effective? Why or why not?
Detention
Youth taken into secure custody at the time of arrest are entitled to judicial review of the detention decision within a statutory time period. Depending on the jurisdiction and characteristics of the case, the length of detention may range from several hours to several months. Many professionals view the detention decision as the most significant point in a case. Detention subjects the youth to potential physical and emotional harm. It also restricts the youth’s ability to assist in his or her defense and to demonstrate an ability to act appropriately in the community.
Unfortunately, youth with disabilities are detained disproportionately (Leone et al., 1995). Experts posit that one reason for this is that many youths with disabilities lack the communication and social skills to make a good presentation to arresting officers or intake probation officers. Behavior interpreted as hostile, impulsive, unconcerned, or otherwise inappropriate may reflect the youth’s disability. This is another reason why it is important to establish the existence of special education needs or suspected disabilities early in the proceedings. Juvenile justice professionals must be sensitive to the impact of disabilities on case presentation at this initial stage and work to dispel inaccurate first impressions at the detention hearing.
In some cases, it may be appropriate for the court to order the youth’s release to avoid disrupting special education services. This is particularly true if adjustments in supervision (e.g., modification of the IEP or behavioral intervention plans) may reduce the likelihood of further misbehavior pending the jurisdictional hearing. Similarly, if there are early indications that a special education evaluation is needed, it may be important for the youth to remain in the community to facilitate the evaluation. Many jurisdictions have home detention programs that facilitate this type of release by imposing curfews or other restrictions on liberty that allow the youth to live at home and attend school pending the outcome of the delinquency proceedings.
Education
Education may be the single most important service the juvenile justice system can offer young offenders in its efforts to rehabilitate them and equip them for success. School success alone may not stop delinquency, but without it, troubled youth have a much harder time (Beyer, Opalack, and Puritz, 1988). When special education needs are evident, they should be an essential part of the social study report prepared by the probation department to guide the court in making its disposition order. Moreover, juvenile justice professionals should coordinate disposition planning with education professionals to avoid conflict and to take advantage of the rich evaluation resources and services available through IDEA.
The resulting disposition order should reflect the court’s review of special education evaluations and the goals, objectives, and services to be provided under the IEP. If the youth is to be placed out of the home, the court should demand specific assurance that the facility will meet the youth’s educational needs under IDEA. The juvenile court should also use its disposition powers to ensure special education evaluation and placement for previously unidentified youth who show indications of having a disability.
In deciding whether or where to place a youth with a disability, it is also important for the court to understand the impact of the disability on behavior. Youth with attention deficit disorder (ADD), for example, commonly act impulsively, fail to anticipate consequences, engage in dangerous activities, have difficulty with delayed gratification, have a low frustration threshold, and have difficulty listening to or following instructions. They may begin to associate with delinquents or self-medicate through drugs and alcohol because they are rejected by others. Proper medication has a dramatic effect in helping many of these youth control their behavior, and a variety of professionals are skilled in treating ADD in medical, psychiatric, or educational settings (Logan, 1992). Unless the characteristics of ADD and the existence of effective interventions are recognized, youth with this disability stand a good chance of being treated harshly, often through incarceration, based on the outward manifestations of their disability. Juvenile justice professionals should respond appropriately to evidence of such disabilities by ensuring that appropriate medical, mental health, and other services are provided.
Juvenile justice professionals also must learn to recognize potential problems for youth with certain disabilities in particular settings, so as not to set the youth up for failure. This does not mean that juvenile justice professionals need to become diagnosticians or clinicians. However, they should consult with education, mental health, and medical professionals. It is important to seek professional advice about the kinds of settings in which the youth can function best and the kinds of settings most likely to lead to negative behavior. For example, a youth with an emotional disturbance may not be able to function in the large dormitory setting typical of some institutions. Such youth may feel especially vulnerable because of past physical or sexual abuse or may simply suffer from over stimulation in an open setting. They may require a setting in which external stimuli are reduced to the greatest extent possible and intensive one-on-one supervision is provided. Youth with other disabilities may need programs that minimize isolation and emphasize participation in group activities.