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We studied adaptive thermotolerance in replicate populations of Drosophila melanogaster artificially selected for high and low knockdown temperature (TKD), the upper temperature at which flies can no longer remain upright or locomote effectively. Responses to selection have generated High TKD populations capable of maintaining locomotor function at ∼40°C, and Low TKD populations with TKD of∼35°C. We examined inducible knockdown thermotolerance, as well as inducible thermal survivorship, following a pretreatment heat-shock (known to induce heat-shock proteins) for males and females from the TKD selected lines. Both selection for knockdown and sex influenced inducible knockdown thermotolerance, whereas inducible thermal survivorship was influenced only by sex, and not by selection. Overall, our findings suggest that the relationships between basal and inducible thermotolerance are contingent upon the methods used to gauge thermotolerance,as well as the sex of the flies. Finally, we compared temporal profiles of the combined expression of two major heat-shock proteins, HSC70 and HSP70, during heat stress among the females and males from the selected TKD lines. The temporal profiles of the proteins differed between High and Low TKD females, suggesting divergence of the heat-shock response. We discuss a possible mechanism that may lead to the heat-shock protein patterns observed in the selected females.

Most organisms in nature are exposed chronically to fluctuating environmental conditions, and as a consequence, have evolved stress response networks that activate when ambient conditions approach stressful levels. One means by which virtually all organisms combat stress is through the heat-shock response (Lindquist and Craig,1988; Morimoto et al.,1994; Feder and Hofmann,1999). Heat-shock proteins (HSPs) are principle constituents of this response and typically function as molecular chaperones, which assist in maintaining cellular protein structure. During thermal stress, members of the HSP70 family mitigate cellular damage in Drosophila (Georgopoulis and Welch, 1993; Parsell and Lindquist,1993; Hartl, 1996; Sørensen et al., 2003; Gong and Golic, 2006). HSC70 and HSP70 are the most abundantly expressed members of this family, and both have been implicated in tolerance to hyperthermia. HSC70 is expressed constitutively and is linked to basal thermotolerance, whereas the expression of HSP70 is heat-induced and is linked to acquired (inducible) thermotolerance(Lindquist, 1980; Velaquez,1983; Palter et al., 1986; Leung et al., 1996; Feder and Hofmann, 1999; Sung and Guy, 2003). The role of HSP68, the third principle HSP70 family member, in thermotolerance is not well-established (McColl et al.,1996).

Drosophilids are a model system used to study the contribution of the heat-shock response to thermotolerance. Drosophilids are thermoconformers,that is, their small body size demands that body temperature conform to ambient temperatures. In Drosophila, exposure to stressful temperature generally induces synthesis of HSP70 (e.g. Lindquist, 1981; Welte et al., 1993; Krebs and Loeschcke, 1994; Feder et al., 1996; Sørensen et al., 2003). HSP70 induction comes with costs and has been shown to adversely impact development and reduce reproductive capacity. In D. melanogaster,HSP70 induction is implicated in reduced fecundity(Krebs and Loeschcke, 1994),and HSP70 overexpression is associated with increased larval mortality(Krebs and Feder, 1997b),retarded growth (Feder et al.,1992), and reduced egg hatching(Silbermann and Tatar,2000).

While HSP70 expression in D. melanogaster incurs high costs,induced thermotolerance due to HSP70 accumulation is well-established (e.g. Welte et al., 1993; Feder et al., 1996; Feder and Krebs, 1998; Krebs and Feder, 1997a). Juvenile drosophilids are confined to their feeding environment and,therefore, may be unable to seek refuge from thermal stress. In contrast,adults can fly and may rely on behavior to escape. Despite a capacity for escaping heat stress via flight, temperatures in nature can increase quite suddenly (Dahlgaard and Loeschcke,1997), and adults may find themselves in transit during stress-inducing temperature transitions. In these situations, the maintenance of flight allows adults to seek a thermal refuge. Induction of HSP70 appears to protect adult D. melanogaster against locomotor dysfunction during thermal stress (Roberts et al.,2003; Klose and Robertson,2004), suggesting that HSPs have ecologically relevance for adults, as well as juveniles (see also Michalak et al., 2001; Sørensen et al., 2003; Gong and Golic, 2006).

Relatively few studies have examined the impact of HSPs on locomotor function in insects, but one of the few is particularly germane to our current work (Newman et al., 2004). In the study, the effect of HSPs on locomotion was compared between a desert and a temperate species of Drosophila. Thermosensitivity of locomotor behavior in larvae of the desert species (D. arizonae) was reduced,relative to that of the temperate species (D. melanogaster). The upper temperature limit for locomotor function in D. arizonaeexceeded that of D. melanogaster by 6°C. Conversely, exposure to 40°C was associated with improved synaptic function at neuromuscular junctions only in D. melanogaster, not in D. arizonae. The high basal thermotolerance in D. arizonae was correlated with high levels of constitutively expressed HSC70, whereas the low inducible thermotolerance of this species was correlated with the absence of inducible HSP70 at 40°C. In general, desert species of Drosophila, as well as species' strains from hot climates, appear to have high constitutive levels of HSC70 and relatively high temperature set-points for HSP70 induction(Sørensen et al., 2001; Zatsepina et al., 2001). This profile of high constitutive levels coupled with high temperature induction of HSPs has also been observed in desert lizards(Ulmasov et al., 1992; Zatsepina et al., 2000).

In this study, we used artificial selection to generate replicate populations of Drosophila melanogaster capable of locomotor function at high temperatures (∼40°C). The adults from these populations were selected for high knockdown temperature (TKD), the upper temperature at which flies are unable to locomote effectively or remain upright (Gilchrist and Huey,1999). The High TKD flies display enhanced thermotolerance, relative to the Low TKD populations selected for low knockdown temperature. In nature, populations of Drosophila melanogaster exhibit clinal variation in knockdown time and temperature along the eastern coast of Australia, with flies from low latitudes having a significantly higher knockdown than those from high latitudes (Hoffmann et al.,2002) (G.W.G. and R. B. Huey, unpublished). Adaptive changes in thermotolerance may be mediated by modifications within the heat-shock response, which has been demonstrated in a variety of natural and experimental Drosophila populations (McColl et al., 1996; Krebs and Feder,1997a; Dahlgaard et al.,1998; Feder and Krebs,1998; Lerman and Feder,2001; Michalak et al.,2001; Sørensen et al.,2001; Zatsepina et al.,2001; Bettencourt et al.,2002).

We propose that the knockdown lines provide a useful model of adaptation to thermal stress, and that locomotor performance at high temperature in the High TKD lines is supported, in part, by modifications within the heat-shock response. Furthermore, selection for high or low knockdown temperature has altered basal TKD thermotolerance in our selected lines, and we argue that inducible thermotolerance may be influenced as well. More specifically, we propose that the HSP profile of the High TKD flies may resemble that of high-temperature adapted Drosophila. We predict that females and males from the High TKD lines, when compared to the same sex from the Low TKD lines, will have: (i) higher constitutive levels of HSC70, and (ii) a temperature set-point for HSP70 induction that is higher than 36°C, a temperature that induces HSP70 synthesis in adults from temperate populations (Zatsepina et al.,2001). Assuming that the temperature set-point for HSP70 induction has shifted upward in the High TKD lines and is>36°C, we predict that, following a 36°C pre-treatment, (iii)locomotor capacity during thermal stress will be improved in Low TKD lines (and Control lines) only, and (iv) inducible thermotolerance, indicated by enhanced thermal survival, will be evident in Low TKD lines (and Control lines) and absent in High TKD lines.

### Establishment of artificially selected fly lines

The founding population of Drosophila melanogaster used in this study was established in 1991 by combining ∼1000 isofemale lines collected by L. Harshmann and M. Turelli in Escalon, California, USA. In April 1992, a subset of this population (∼1000 females) was used to generate a large number of offspring, which were transferred to population cages, each containing 2000–3000 adults (for details, see Gilchrist and Huey, 1999). The flies were maintained in discrete 2-week generations at 25°C with a 12 h:12 h L:D photoperiod. In 1993–1994, six High Knockdown lines (High TKD'), three Low knockdown lines (Low TKD'), and six Control lines were established. The lines underwent artificial selection (as described below) for 46 generations, ending in September 1997, at which time they were maintained in discrete 3-week generations without selection. In June 2004, selection on these original lines was resumed and continued for 7 generations.

In January 2005, we combined flies from the original lines within each selection group and use these flies to generate new High TKD, Low TKD and Control lines(Fig. 1). To establish each of the four new High TKD lines (HN1–4), 25 males and 25 females from each of the six original High TKD lines were haphazardly chosen and combined. The same procedure was repeated using flies from the six original Control lines to establish four new Control lines (CN1–4). Each of the four new Low TKD lines (LN1–4) was generated by combining 50 males and 50 females haphazardly chosen from each of the three original Low TKD lines. We generated these new selection lines in order to: (1) reduce inbreeding by crossing the original lines within each treatment group, and (2) create equal numbers of lines within the HN, LN and CN treatment groups. All experiments in this study were done using the new selection lines that had undergone selection for up to 9 generations.

Fig. 1.

A diagrammatic depiction of the selection history of the original and newly established High TKD, Low TKD, and Control lines. Details of the selection history are discussed in Materials and methods.

Fig. 1.

A diagrammatic depiction of the selection history of the original and newly established High TKD, Low TKD, and Control lines. Details of the selection history are discussed in Materials and methods.

### Measuring knockdown temperature

The knockdown protocol is detailed elsewhere(Huey et al., 1992; Gilchrist and Huey, 1999). Approximately 1000 flies from each line are poured into the inner tube of a water-jacketed Weber column (Weber,1988). The water jacket is heated by a Haake DC-10 immersion heater (Haake-Buchler Inc., Paramus, NJ, USA) in a 26 liter water bath and contains a copper tube through which air is heated to the same temperature and forced through the column. The flies are poured into the inner tube when the air core temperature is 30°C. Mesh baffles within the inner tube provide places for the flies to cling during the experiment. The water temperature is ramped from 30°C up to 50°C over ∼50 min; the average rate of change is ∼0.4°C min–1. TKD is that temperature at which a fly can no longer locomote or cling to the baffles or column walls; they fall out of the column. As they fall out, the flies are fractionated at 0.5°C intervals between 32°C to 46°C. Recovery is nearly instantaneous (Gilchrist and Huey,1999). The knockdown flies are then separated by sex, counted, and the distribution of knockdown temperatures recorded separately for males and females from each line.

### Selection and maintenance protocols of fly lines

To maintain each HN line, ∼30% of the flies with the highest TKD are selected and retained for breeding. For each LN line, ∼30% of the flies whose TKD generally ranges from 35.5–37°C are selected and retained for breeding. After each Control line is run through the KD column, ∼30% of all the flies are haphazardly chosen for retention and breeding. Following a round of selection,the selected flies from each line (∼300 individuals with approximately equal number of males and females) are divided into two groups of ∼150 flies, each of which is placed into a bottle containing 30 ml cornmeal–molasses–yeast–agar–tegosept medium sprinkled with yeast. The flies are maintained at 25°C for 5–6 days following knockdown to ensure mating between selected males and females(Gilchrist and Huey, 1999). They are then transferred to fresh food sprinkled with yeast and allowed to oviposit for ∼24 h. The eggs are transferred to fresh food vials at a density of ∼50 eggs/vial. Twenty-eight vials of eggs are collected from each line. Flies develop and mature at 25°C (12 h:12 h L:D) for 13 days,at which time the adults are transferred to fresh food bottles sprinkled with yeast in preparation for the next knockdown experiment, which is run on days 14 and 15 following egg collection. (The selection and maintenance protocols have remained consistent for the original and the new knockdown lines.)

### Induction of HSP70 and knockdown temperature

Flies from each selection line were removed from selection for two generations prior to egg collection to minimize cross-generational effects. Twenty vials of eggs (∼50 eggs/vial) were collected from each CN, LN and HN line. At 14 days following egg collection, flies from four food vials were transferred into a single food bottle; this was repeated until we had collected five food bottles from each line, each containing ∼200 flies. Immediately prior to the pretreatment (36°C for 1 h, followed by 25°C for 1 h), flies in each bottle were transferred to an empty food bottle containing a piece of moistened filter paper (∼1 cm2). Immediately following pretreatment, the flies were run through the knockdown column, and the distribution for knockdown temperatures was recorded (as described above). The distribution of TKD of flies not subjected to the pretreatment was also recorded within the same generation for all CN, LN and HN lines.

### Basal and induced thermal survival

To minimize cross-generational effects, all flies were removed from selection for one generation prior to egg collection. Eggs (∼50 eggs/vial;28 vials/line) were collected from all CN, LN and HN lines. Each fly line was treated according to the following protocol. Thirteen days following egg collection, adults were CO2-anesthetized and combined. Two groups of ∼180 flies were generated haphazardly, and each group was transferred to a fresh food bottle. Three days later, the flies in both bottles were CO2-anesthetized, combined, separated according to sex, and distributed haphazardly into two groups of food vials: five vials with ∼15 males in each (Group 1) and five additional food vials with ∼15 males in each (Group 2). The same collection scheme was repeated for the females. All flies were allowed to recover from CO2-anesthesia for ∼36 h. Groups 1 and 2 were treated according to the following protocols.

#### (1) Pretreatment+heat stress

The flies from each food vial in Group 1 were transferred to a 12 mm×75 mm glass culture tube containing 1 ml of 2% agar. Culture tubes were capped (pin holes were made in caps), placed into a 36°C water bath for 1 h, and then held at 25°C for 1 h. Immediately following this pretreatment, the flies were heat-shocked at 38.5°C for 1 h, held at 25°C for 24 h, and then scored for survivorship.

#### (2) Heat-stress only

Flies in Group 2 were treated exactly as described above for Group 1,except the pretreatment (i.e. 36°C for 1 h, and then 25°C for 1 h) was omitted. Proportional survivorship of flies in each culture tube was determined by dividing the number of flies alive in a tube at 24 h by the number of flies initially placed in the tube.

### The heat-shock response: heat-stress treatment and sample preparation

Two major HSP70 family members, HSC70 and HSP70, were assayed separately in males and females following exposure to 36°C at 0 min, 10 min, 20 min, 30 min or 60 min. Samples were prepared from each selected line(HN1–4 and LN1–4) according to the following protocol. On day 13 or 14 following the egg stage, adults were CO2-anesthetized and separated according to sex. Six groups of 20–25 males/group, and six groups of 20–25 females/group, were each placed in fresh food vials and allowed to recover from anesthesia for∼24 h. One sample (i.e. 20–25 flies) from each sex was transferred to a 1.5 ml screw-top tube, quick-frozen and stored at –80°C. The remaining samples (5 groups from each sex) were each transferred to an empty food bottle containing moistened filter paper (∼2 cm2) and placed in a 36°C water bath. One sample from each sex was removed from the water bath after 10 min, 20 min, 30 min or 60 min. Upon removal, each sample was transferred immediately to a 1.5 ml screw-top tube, quick-frozen and stored at –80°C. [Quick-freezing adults at –70°C does not affect HSP70 levels compared to flash-freezing adults in liquid nitrogen(Dahlgaard et al., 1998).]

Each sample was prepared for protein gel electrophoresis (12% SDS-PAGE) by homogenizing the flies in 200–400 μl of ice-cold, 2 mmol l–1 PMSF (phenylmethylsulphonyl fluoride, protease inhibitor,MP Biomedicals, Inc., Aurora, OH, USA) in 1× phosphate-buffered saline(PBS) with a hand-held Kontes homogenizer (Kimble/Kontes, Vineland, NJ, USA). Homogenized samples were centrifuged at 14 000 g for 30 min at 4°C. Samples of the supernatant (40–60 μl) were transferred to microcentrifuge tubes, quick-frozen and stored at –80°C.

#### Western blot analysis

Total protein concentration of the samples (supernatants) was measured using a Bradford Protein Assay according to the manufacturer's instructions(Bio-Rad Laboratories, Hercules, CA, USA). Samples were denatured by heating at 95°C for 5 min in SDS sample buffer (final concentrations: 12.5%glycerol, 0.5% Bromophenol Blue, 1.25% sodium dodecylsulfate, 0.5 mol l–1 Tris-Cl, pH 6.8, 1.25% β-mercaptoethanol). 30–50 μg of protein was loaded into each well for electrophoresis,and 5 ng of HSP70 protein (Product # NSP-555, Stressgen Biotechnologies Corp.,San Diego, CA, USA) was included as a standard. Proteins were separated on 12%polyacrylamide gels (mini-Protean II, Bio-Rad Laboratories) for ∼45 min at 180 V, according to standard methods. Separated proteins were transferred onto a PVDF (polyvinylidene difluoride) membrane (Hybond™-P, GE Healthcare Life Sciences, Waukesha, WI, USA; Protean II system, Bio-Rad Laboratories). Following transfer, blots were incubated in 5% non-fat dried milk in phosphate-buffered saline (1.9 mmol l–1NaH2PO4. H2O, 8.1 mmol l–1Na2HPO4.7H2O, 137 mmol l–1NaCl, 2.6 mmol l–1 KCl, pH 7.4) + 0.1% Tween®20 (PBS-T) either overnight at 4°C or for 1 h at 23°C, followed by a 1-h incubation using two primary antibodies simultaneously: (1) rabbit anti-rat HSC70/HSP70 polyclonal antibody (in whole rabbit antiserum) against amino acid residues 446–641 (Stressgen Biotechnologies Corp., San Diego,CA, USA, Product # SPA-757; 1:100 000 in blocking solution); and (2) rabbit anti-actin polyclonal IgG antibody against amino acid residues 20–33 of invertebrate actin (Sigma, St Louis, MO, USA, Product# A5060, 2.46 μg ml–1 in blocking solution). Following two 10-min washes in PBS-T, blots were incubated for 1 h with the secondary antibody: goat anti-rabbit IgG polyclonal antibody conjugated to HRP (horseradish peroxidase;Stressgen Biotechnologies Corp., Product #SAB-300, 1:10 000 in blocking solution). Blots were washed 3× for 10 min each wash in PBS-T. Chemifluorescence of immunoreactions was detected using ECL Plus™Western Blotting Detection Reagents according to the manufacturer's instructions (GE Healthcare Life Sciences). Heat-shock proteins and actin were quantified with a Molecular Dynamic STORM Phosphorimager (GE Healthcare Life Sciences) and software, ImageQuant, version.5.2. Quantification of actin was used to normalize the protein concentrations of the samples.

#### Statistical analyses

All statistical analyses were performed using R, version 2.2.1(R Development Core Team,2006). Estimation of the parameters for the unimodal and bimodal distributions for TKD was by restricted maximum likelihood(Gilchrist and Huey, 1999). The hypothesis that a bimodal distribution provided a better description of the data than the unimodal distribution was tested using a log-likelihood ratio test. Additional statistical analyses, all performed in R(version 2.2.1), are described in the Results.

### Response to knockdown selection

All the original lines displayed a bimodal distribution for TKD at the onset of selection, as detailed elsewhere(Gilchrist and Huey, 1999). Bimodality has been observed in ∼50 populations of D. melanogaster for which TKD has been measured (G.W.G.,unpublished), except in the TKD selected lines. While the bimodal distribution for TKD was retained in the original Control lines, the original High TKD lines lost most of the lower mode and displayed an increase in the mean of the upper mode. Conversely, the original Low TKD lines lost most of the upper mode and displayed a decrease in the mean of the lower mode.

Fig. 2.

Histograms of TKD for the newly established LN, CN and HN lines in generations 1 (top) and 12 (bottom). At generation 12, the flies had undergone 9 generations of knockdown selection. The error bars show s.e.m. values for groups of flies fractionated at 0.5°C intervals. Each LN, CN and HN group comprises four replicate populations. During knockdown, ∼1000 flies from each line are assayed for TKD; thus, each graph represents TKD frequency distribution for ∼4000 flies. The fitted line on each graph shows the maximum likelihood fit for a bimodal distribution. The fitted parameters for the distributions are listed in Table 1.

Fig. 2.

Histograms of TKD for the newly established LN, CN and HN lines in generations 1 (top) and 12 (bottom). At generation 12, the flies had undergone 9 generations of knockdown selection. The error bars show s.e.m. values for groups of flies fractionated at 0.5°C intervals. Each LN, CN and HN group comprises four replicate populations. During knockdown, ∼1000 flies from each line are assayed for TKD; thus, each graph represents TKD frequency distribution for ∼4000 flies. The fitted line on each graph shows the maximum likelihood fit for a bimodal distribution. The fitted parameters for the distributions are listed in Table 1.

The mean distributions for TKD in the newly established lines (i.e. HN, LN and CN) for the 1st and 12th generations are shown in Fig. 2, with parameters given in Table 1. Both the LN and the CN lines show bimodality in Generation 1, after combining the original lines. In contrast, the HN lines are more nearly unimodal, although there was some regression towards a lower mean TKD after combining the original lines. By Generation 12 (after 9 generations of selection: selection was not imposed for 3 of the 12 generations), the new HN and LN lines more closely resembled the original TKD lines at Generation 46(Gilchrist and Huey, 1999). In the LN lines, both females and males have nearly lost the upper mode. The proportions of LN flies that fell under the lower mode ranged from 0.96 to 1.00 (mean=0.98) and 0.83 to 1.00 (mean=0.93) for the females and males,respectively (Table 1). The mean temperature of the lower mode for both sexes was lower in the LN lines(34.7°C) than in the HN lines (37.1°C). Interestingly, the HN lines somewhat increased in bimodality for TKD. Nonetheless, a high proportion of the HN flies fell under the upper mode (females ranged from 0.47 to 0.80, mean=0.66; males ranged from 0.83 to 1.00, mean=0.90). The mean temperature of the upper mode for both sexes was higher in the HN lines(41.1°C) than in the LN lines (38.1°C).

Table 1.

Maximum likelihood estimates for unimodal and bimodal KD temperature(TKD) distributions for Control lines, High KD lines and Low KD lines in generation 12

TreatmentSexReplicateMu (Su)PlowMlow (Slow)Mhigh (Shigh)GP
Control 35.23 (1.35) 1.00 35.23 (1.35) NA 0.00 1.000
34.85 (1.62) 0.86 34.45 (1.04) 37.27 (2.30) 161.50 0.000
35.86 (1.89) 0.83 35.17 (1.10) 39.13 (1.33) 176.18 0.000
35.51 (1.88) 0.85 34.93 (1.27) 38.87 (1.24) 79.38 0.000
Control 36.40 (1.88) 0.57 35.33 (0.82) 37.84 (1.94) 162.07 0.000
36.19 (2.19) 0.73 35.37 (1.29) 38.44 (2.54) 124.64 0.000
37.29 (2.40) 0.60 35.72 (1.18) 39.67 (1.73) 104.91 0.000
36.78 (2.71) 0.62 35.30 (1.30) 39.19 (2.68) 142.39 0.000
High 40.40 (2.15) 0.20 37.08 (1.18) 41.23 (1.39) 97.33 0.000
39.24 (2.21) 0.41 37.22 (2.06) 40.67 (0.60) 328.92 0.000
39.57 (2.55) 0.53 38.29 (2.93) 40.99 (0.61) 295.80 0.000
39.79 (2.12) 0.23 36.78 (1.12) 40.69 (1.41) 68.39 0.000
High 40.80 (1.98) 0.00 NA 40.80 (1.98) 0.00 1.000
40.38 (1.90) 0.17 36.92 (1.37) 41.10 (0.97) 215.00 0.000
40.96 (2.19) 0.12 36.39 (1.11) 41.57 (1.45) 144.42 0.000
40.77 (1.96) 0.12 37.21 (1.28) 41.27 (1.46) 48.94 0.000
Low 34.42 (1.17) 0.99 34.38 (1.10) 38.86 (0.65) 28.59 0.000
34.33 (1.05) 1.00 34.33 (1.03) 38.50 (0.00) 63.59 0.000
34.77 (1.18) 0.96 34.64 (0.95) 38.08 (1.71) 105.32 0.000
34.47 (1.07) 0.98 34.41 (0.93) 38.68 (1.13) 79.63 0.000
Low 35.21 (1.51) 0.90 34.86 (1.06) 38.38 (1.34) 118.39 0.000
34.86 (1.01) 0.97 34.77 (0.89) 37.57 (0.66) 28.01 0.000
35.27 (1.42) 1.00 35.27 (1.42) NA 0.00 1.000
35.10 (1.33) 0.83 34.74 (0.78) 36.87 (1.91) 220.11 0.000
TreatmentSexReplicateMu (Su)PlowMlow (Slow)Mhigh (Shigh)GP
Control 35.23 (1.35) 1.00 35.23 (1.35) NA 0.00 1.000
34.85 (1.62) 0.86 34.45 (1.04) 37.27 (2.30) 161.50 0.000
35.86 (1.89) 0.83 35.17 (1.10) 39.13 (1.33) 176.18 0.000
35.51 (1.88) 0.85 34.93 (1.27) 38.87 (1.24) 79.38 0.000
Control 36.40 (1.88) 0.57 35.33 (0.82) 37.84 (1.94) 162.07 0.000
36.19 (2.19) 0.73 35.37 (1.29) 38.44 (2.54) 124.64 0.000
37.29 (2.40) 0.60 35.72 (1.18) 39.67 (1.73) 104.91 0.000
36.78 (2.71) 0.62 35.30 (1.30) 39.19 (2.68) 142.39 0.000
High 40.40 (2.15) 0.20 37.08 (1.18) 41.23 (1.39) 97.33 0.000
39.24 (2.21) 0.41 37.22 (2.06) 40.67 (0.60) 328.92 0.000
39.57 (2.55) 0.53 38.29 (2.93) 40.99 (0.61) 295.80 0.000
39.79 (2.12) 0.23 36.78 (1.12) 40.69 (1.41) 68.39 0.000
High 40.80 (1.98) 0.00 NA 40.80 (1.98) 0.00 1.000
40.38 (1.90) 0.17 36.92 (1.37) 41.10 (0.97) 215.00 0.000
40.96 (2.19) 0.12 36.39 (1.11) 41.57 (1.45) 144.42 0.000
40.77 (1.96) 0.12 37.21 (1.28) 41.27 (1.46) 48.94 0.000
Low 34.42 (1.17) 0.99 34.38 (1.10) 38.86 (0.65) 28.59 0.000
34.33 (1.05) 1.00 34.33 (1.03) 38.50 (0.00) 63.59 0.000
34.77 (1.18) 0.96 34.64 (0.95) 38.08 (1.71) 105.32 0.000
34.47 (1.07) 0.98 34.41 (0.93) 38.68 (1.13) 79.63 0.000
Low 35.21 (1.51) 0.90 34.86 (1.06) 38.38 (1.34) 118.39 0.000
34.86 (1.01) 0.97 34.77 (0.89) 37.57 (0.66) 28.01 0.000
35.27 (1.42) 1.00 35.27 (1.42) NA 0.00 1.000
35.10 (1.33) 0.83 34.74 (0.78) 36.87 (1.91) 220.11 0.000

A significant result for the log-likelihood test suggests that the bimodal distribution provides a better description of TKD. Mu and Su are maximum likelihood estimates (MLEs) for means and standard deviations (s.d.) assuming a unimodal model. Mlow and Slow are the mean and s.d. of the low mode and Mhigh and Shigh are the mean and s.d. of the high mode, assuming a bimodal distribution. Plow is the proportion of the flies that fall under the lower mode.

High, high KD lines; and Low, low KD lines; G, G statistic.

### Shifts in knockdown temperature following pretreatment heat-shock

The TKD distribution of males and females of the CN, HN and LN lines was measured immediately following pretreatment and was compared to TKD distribution for flies that were not pretreated(Fig. 3). Paired t-tests were used to compare mean TKD of pretreated flies with mean TKD of non-pretreated flies for each sex from each CN, HN and LN group (e.g. LN females were compared only with non-pretreated LN females). The only group showing a significant increase in TKD following the pretreatment was the LN females(P=0.0236). Both females and males from all HN lines had a significant drop in TKD following the pretreatment(P=0.0002; P=0.0007, respectively). Control males and females, as well as LN males, had no significant change in TKD distribution in response to the pretreatment.

Fig. 3.

The effect of a heat-shock pretreatment on TKD. Each bar represents the difference (in °C) between the mean TKD of pretreated flies and the mean TKD of non-pretreated flies for the CN1–4, LN1–4, and HN1–4lines. The values for the females are shown at the top, males at the bottom. Mean difference in TKD for females (and for males) from each line was estimated using ∼500 pretreated flies and ∼500 non-pretreated flies. Black bars, values for CN1–4; blue bars, LN1–4; red bars, HN1–4.

Fig. 3.

The effect of a heat-shock pretreatment on TKD. Each bar represents the difference (in °C) between the mean TKD of pretreated flies and the mean TKD of non-pretreated flies for the CN1–4, LN1–4, and HN1–4lines. The values for the females are shown at the top, males at the bottom. Mean difference in TKD for females (and for males) from each line was estimated using ∼500 pretreated flies and ∼500 non-pretreated flies. Black bars, values for CN1–4; blue bars, LN1–4; red bars, HN1–4.

Fig. 4.

Pretreatment of flies (heat hardening) increases survivorship, measured by proportion of flies that are alive 24 h after thermal stress. HS=38.5°C heat shock for 1 h, with no pretreatment. Pre+HS=36°C heat hardening pretreatment for 1 h followed by a resting period (25°C for 1 h) prior to heat shock at 38.5°C for 1 h. Each data point represents the mean survivorship of ∼300 males or females (i.e. ∼75 flies from each of the four CN, LN or HN lines). Values are means ± s.e.m. Red lines depict change in survivorship in HN flies, with red filled triangles (pointing upward) for females; open triangles, males. Blue lines depict change in survivorship in LN flies, with blue filled triangles (pointing downward) for females; open triangles, males. Black lines depict changes in survivorship in the CN flies, with black filled circles for females; open circles, males.

Fig. 4.

Pretreatment of flies (heat hardening) increases survivorship, measured by proportion of flies that are alive 24 h after thermal stress. HS=38.5°C heat shock for 1 h, with no pretreatment. Pre+HS=36°C heat hardening pretreatment for 1 h followed by a resting period (25°C for 1 h) prior to heat shock at 38.5°C for 1 h. Each data point represents the mean survivorship of ∼300 males or females (i.e. ∼75 flies from each of the four CN, LN or HN lines). Values are means ± s.e.m. Red lines depict change in survivorship in HN flies, with red filled triangles (pointing upward) for females; open triangles, males. Blue lines depict change in survivorship in LN flies, with blue filled triangles (pointing downward) for females; open triangles, males. Black lines depict changes in survivorship in the CN flies, with black filled circles for females; open circles, males.

### Pretreatment heat-shock induces thermal survival

Males and females from all CN, LN and HN lines demonstrated induced thermal survivorship, with the females showing the highest induced (and basal)survivorship (Fig. 4). We used generalized linear modeling for a binomially distributed response variable and a logit link function to test for differences in thermal survivorship between non-pretreated and pretreated flies. Because the females had much higher survivorship overall, the data for males and females were analyzed separately. The analyses indicated that: (1) the pretreatment significantly increased survivorship in both sexes (D[1,118]=349.65, P=5.050e–78; and D[1,118]=906.94, P=3.047e–199,for females and males, respectively); and (2) there was a weak selection× treatment interaction in both sexes(D[2,105]=6.51, P=0.04; D[2,105]=6.22, P=0.04, for females and males,respectively). Most of the variation among the selected lines was in the non-pretreated group (Fig. 4):the HN females had a slightly higher survivorship than the CN or LN females,and the CN males had a slightly higher survivorship than the LN or HN males. The survivorship of pretreated females from all lines was very similar;likewise for the males.

### Profiles of HSC70/HSP70 protein after heat-stress

HSC70/HSP70 protein levels were measured at multiple time intervals in TKD selected females and males held at 36°C. HSC70/HSP70 values from the samples on each gel were normalized using relative proportions of actin from the same samples. We used linear regressions to test for a functional relationship between actin proportions and the time intervals(data not shown). Actin proportions were log transformed prior to analyses. The relative proportions of actin did not change significantly over time in both the HN and LN groups (F[1,115]=3.21, P=0.0758 and F[1,120]=1.366, P=0.2448,respectively).

Fig. 5.

Total HSC70/HSP70 protein content prior to (0 min) and at 10, 20, 30 and 60 min following a 36°C heat stress in female and male flies. Results are expressed relative to 5 ng HSP70 protein included as a standard (see Materials and methods). The data were adjusted for block effects of running western blots in groups. Solid red lines and red filled triangles, values for the HN females; solid blue lines and blue filled triangles, LN females; broken red lines and open red triangles, HN males; broken blue lines and open blue triangles, LN males. Values are mean ± s.e.m. of 8 samples (2 samples from each HN1–4 or LN1–4 line).

Fig. 5.

Total HSC70/HSP70 protein content prior to (0 min) and at 10, 20, 30 and 60 min following a 36°C heat stress in female and male flies. Results are expressed relative to 5 ng HSP70 protein included as a standard (see Materials and methods). The data were adjusted for block effects of running western blots in groups. Solid red lines and red filled triangles, values for the HN females; solid blue lines and blue filled triangles, LN females; broken red lines and open red triangles, HN males; broken blue lines and open blue triangles, LN males. Values are mean ± s.e.m. of 8 samples (2 samples from each HN1–4 or LN1–4 line).

The data were analyzed for the effect of sex and selection treatment on HSP levels over time, with the time of heat exposure treated as an orthogonal polynomial. (The data were adjusted for block effects of running western blots in groups.) A third-order orthogonal polynomial provided only a marginally better fit (F[64,68]=2.2247, P=0.07612) than a second-order, so we used the second-order to avoid over-fitting the model. We tested for homogeneity of the orthogonal polynomial coefficients among the sex× selection groups and could not reject the null hypothesis(homogeneity: F[2,68]=0.93, P=0.401). We then used an analysis of covariance, fitting common second-order orthogonal polynomial coefficients to the sex × selection groups, and tested for homogeneity among the intercepts. A significant interaction between sex and selection (F[1,74]=4.84, P=0.031) suggests that males and females responded differently to thermal stress. The HN females have significantly higher levels of the HSPs than LN females (Tukey HSD, P=0.0175), whereas there is no significant difference in protein levels between the HN and LN males (Tukey HSD, P=0.9991).

The data were also analyzed for the effect of selection treatment on HSP levels over time separately for the sexes. We used a REML (restricted maximum likelihood estimation) fit of a mixed-effects model, with line within selection treatment' treated as a random factor. The results from these analyses show concurrence with those from the analyses described above: the females from the LN and HN lines differed in HSP content over time (Likelihood Ratio Test: χ2[1]=6.36,P=0.001165), while the LN and HN males showed no significant difference(LRT: χ2[1]=0.76, P=0.383).

TKD selection did not affect constitutive levels of HSC70, as shown at 0 min' for both sexes(Fig. 5). The only difference in constitutive HSC70 was between the sexes: the females had ∼40% more HSC70 than the males. In all groups, protein levels dropped initially and then began to climb after 20 min (10 min in HN males). By 60 min, the HSP content had surpassed constitutive levels in all groups, suggesting that HSP70 induction had occurred (Palter et al.,1986). The HN and LN males showed no statistically significant difference in HSC70/HSP70 content at 60 min, although the average level in HN males surpassed that of the LN males by ∼25%. The HN females, relative to the LN females, had significantly higher HSP levels at 60 min.

### Responses to selection for high and low resistance to TKD

Through artificial selection for high TKD, we generated replicate populations of D. melanogaster that maintain locomotor function at ∼40°C (Fig. 2 and Table 1). Selection for low TKD generated replicate populations with a mean TKD of ∼35°C, resembling TKD of the Control lines (mean TKD:∼36°C). As a consequence of selection, the bimodal distribution for TKD consistently observed in many populations of D. melanogaster was largely diminished in both the High and Low TKD lines (HN and LN, respectively). In the LN lines, both sexes generally lost the upper mode, while in the HN lines both sexes tended to lose the lower mode. Loss of the lower mode in HN males is particularly striking: only ∼10% of the males fell under the lower mode. More of the HN females (∼33%) fell under the lower mode, suggesting that despite strong selection for high TKD, the lower mode in HN females was moderately less responsive to selection. The biological significance of the widespread bimodal distribution for TKD is unclear, but its persistence in natural populations of D. melanogaster suggests that there may be an adaptive advantage to retaining both modes and that the fixation of either mode may entail some significant trade-off in fitness. The bimodality of wild (and unselected) flies may reflect a polymorphism regulating thermotolerance, or a multilocus system for dealing with heat,cold, or levels of heat-shock proteins.

### Influence of pretreatment on TKD

Prior to measuring TKD in this experiment, LN, HN and CN flies were subjected to a pretreatment heat-shock known to induce HSP70 in many temperate populations of Drosophila melanogaster. We predicted that pretreatment would improve knockdown performance in LN (and CN) flies. This prediction proved accurate for LN females only(Fig. 3). During a typical knockdown experiment (no pretreatment), an average LN female is heat-stressed for ∼13 min, during which their HSC70 content drops precipitously(Fig. 5). When the flies are pretreated just prior to measuring knockdown, the HSP profile during knockdown may be quite different due to the heat-shock response elicited by the pretreatment. We propose that the improved knockdown performance in LN females may be due to HSP70 induction, which appears to improve locomotor function in adults from some populations of D. melanogaster(Roberts et al., 2003; Newman et al., 2004; Klose and Robertson, 2004). It is unclear why the pretreatment had no significant effect on knockdown performance in LN males. Our data indicate that HSP70 induction occurs in the males also. This discrepancy underscores the difference in stress responses between males and females.

The pretreatment heat-shock had an adverse effect on TKD in both HN males and females: TKDdropped ≥2°C for most of the HN lines(Fig. 3). We speculate that energetic trade-offs may contribute to the reduction in TKD. During a typical knockdown experiment, HN flies are heat-stressed for ∼25 min at temperatures approaching ≥40°C. During a typical knockdown experiment, HSC70 levels drop and HSP70 induction is initiated (Fig. 5). As discussed above, when flies are pretreated just prior to knockdown, HSP70 has accumulated at the onset of knockdown. As a result, the pretreated HN flies may face strong energetic challenges during knockdown, such as: (1) the ATP-driven activities of HSP70 (and perhaps other molecular chaperones),coupled with (2) rising metabolic demands associated with increasing body temperature. The metabolic rate of HN flies may increase ∼70% during knockdown as temperature reaches TKD (data not shown). We propose that at very high temperatures metabolic trade-offs may perturb cellular processes fundamental to locomotor function in the pretreated HN flies, resulting in lower than normal TKD.

### Basal and induced thermal survivorship

We hypothesized that the induction of thermal survivorship through a pretreatment would be evident in LN (and CN) flies, and not HN flies. Our results indicate that enhanced thermal survivorship was induced in both LN and CN flies, as well as in HN flies (Fig. 4).

All males had low basal thermal survivorship (<10%, on average), with CN males faring slightly better than the others, whereas all females had relatively high basal thermal survivorship (∼65%, on average). High levels of HSC70 in females may contribute to their high basal thermotolerance. The HN females had significantly higher basal survivorship (∼80%) than LN females(∼63%); yet basal' HSC70 (i.e. HSC70 of non-stressed flies) did not differ between the HN and LN females. Thus, the difference in survivorship between the females cannot be ascribed to a dosage-effect of basal' HSC70. When heat-stressed for 60 min at 36°C, the HN females consistently maintained higher levels of the HSPs (Fig. 5), so perhaps the maintenance of high levels of the one (or both)of these proteins results in higher basal thermal survivorship.

### Adaptive change in the heat-shock response

Our findings did not support the hypothesis that the lines selected for high knockdown thermotolerance have relatively high levels of constitutive HSC70. The only difference in constitutive HSC70 was between the sexes:non-stressed females had, on average, ∼40% more HSC70 than the males(Fig. 5). This is not surprising, given that the ovaries and embryonic tissues are enriched with HSC70 (Palter et al., 1986). These findings re-emphasize the importance of quantifying HSC70 levels separately in females and males.

At stressful temperatures in D. melanogaster, the synthesis of many proteins is repressed (Storti et al.,1980; Lindquist,1981), but HSC70 exhibits translational thermotolerance and continues to be synthesized, albeit at lower levels than in non-stressed flies(Palter et al., 1986). For several hours following heat-stress, cognate proteins, including HSC70, are the most abundant non-inducible HSPs. Yet our data indicate that after only a few minutes at 36°C, HSC70 levels of males and females from all TKD lines dropped significantly. After 20–30 min,the HSP levels increased above the nadir, which may reflect some degree of recovery of HSC70, as well as the induction of HSP70.

During the initial 20 min of heat-stress, the LN females showed a significantly greater decline in HSC70 compared to the HN females. This early drop in protein levels represented a ∼35% reduction in the LN females, but only ∼18% in the HN females. One explanation for the significantly steeper drop observed in the LN females is that HSC70 degradation proceeds at a higher rate. HSC70 is a key molecular chaperone that binds to unfolded and otherwise impaired proteins, and returns them to their native conformation or assists in their degradation via the ubiquitination system(Bercovich et al., 1997). Degradation of proteins through this system proceeds by tagging of proteins with ubiquitin followed by degradation by 26S proteasomes(Wickner et al., 1999). Although HSC70 is known to conscribe the ubiquitination system for degradation of its substrates (Bercovich et al.,1997), it is possible that HSC70 itself is also targeted for proteolysis through the same system. In support of this putative mechanism, a recent study showed that a ubiquitin ligase, CHIP (carboxy terminus of HSP70-binding protein), ubiquinates HSC70 in vivo, thus tagging it for proteolysis (Qian et al.,2006) and reducing t1/2 (half life) of HSC70 in vitro. We hypothesize that the heat-shock response in HN females involves blunted degradation of HSC70 through changes in the activity of one or more ubiquitin-proteasome factors.

The differential drop in HSC70 protein during heat-stress may be linked to regulation of HSP70 induction. In Drosophila, HSP70 induction is controlled post-translationally through the activity of heat-shock transcription factor, HSF1 (Jedlicka et al., 1997). In non-stressed cells, HSF1 is an inactive monomer. Following exposure to stress, HSF1 trimers translocate to the nucleus and bind to the heat-shock element (HSE) on the HSP70 promotor, activating transcription. Both induced HSP70 and HSC70 proteins act as negative feedback regulators of HSP70 transcription(Morimoto, 1998) by physically binding to HSF1 and attenuating the heat-shock response through repression of transcription (Shi et al.,1998). Conversely, depletion of HSP70/HSC70 (in conjunction with depletions of HSP40 and HSP90) promotes transcriptional activation(Marcher and Wu, 2001).

The idea that a common stress signal activating HSF is the relief of repression imposed by the HSPs' is not novel(Marcher and Wu, 2001). While others have proposed that such relief of HSP70 repression results from the titration of HSPs by impaired proteins(Lindquist and Craig, 1988; Cotto and Morimoto, 1999), we propose that it may also be a consequence of HSC70 degradation. According to our model, HN females display reduced HSC70 degradation during heat-shock. Because HSC70 protein inhibits the activity of HSF1(Morimoto, 1998), less HSC70 degradation in HN females results in increased ability (at least partially) to inhibit HSF1 activity. This would explain both the smaller drop in HSC70(0–20 min) in HN females, as well as the reduced increase in inducible HSP70 after 20 min. We speculate that retention of HSC70 in the HN females during heat-stress may reduce induction of HSP70, thus protecting reproductive capacity following exposure to high temperatures(Silbermann and Tatar, 2000). We are currently examining this idea.

Lastly, an alternative hypothesis would be that HSP70 induction takes place earlier in the HN females compared to the LN females, obscuring a drastic drop in HSC70. Because our antibody detects both HSC70 and HSP70 protein levels, we were unable to determine when the initiation of HSP70 induction takes place. However, a study by Feder et al. suggests that this is not necessarily a problem (Feder et al., 1997). Feder et al. measured induction of HSP70 in Drosophila cell cultures using ELISA. The authors were unable to detect induced HSP70 protein in cells continuously exposed to 36°C for 15 min, and they detected only minor levels after 30 min. This suggests that in our study the initial drop in protein content is showing degradation of HSC70 solely, and that the observed increase in HSP70 at 30 min is a result of HSP70 induction, perhaps coupled with synthesis of HSC70.

The HSP profile and thermal sensitivities of the High TKD lines did not resemble those previously noted in desert and warm-climate Drosophila, in which high basal thermotolerance is associated with high levels of constitutive HSC70 and the temperature set-point for HSP70 induction is shifted upward. Our findings show a different heat-shock response in the thermotolerant flies, as well as complex patterns of thermotolerance.

We found no evidence that the HSPs studied support locomotor function during a typical knockdown experiment; yet they appear to have an impact when induced prior to knockdown. For example, when HN flies were pretreated just prior to knockdown, TKD was reduced significantly. Although induction of HSP70 may reduce TKD thermotolerance in the HN flies, it appears to increase thermal survivorship. Given these contrasting findings, we surmise that any conclusions concerning the effects of HSP70 induction on thermotolerance' are contingent upon the methods used to assay thermotolerance.

The patterns of thermotolerance in the HN lines discussed above indicate a negative relationship between basal and inducible TKDthermotolerance. Furthermore, high basal TKDthermotolerance in HN flies was not correlated with a similarly high level of inducible thermal survivorship. (Inducible thermal survivorship in the HN males did not differ from that of LN or CN males, which have lower basal TKD thermotolerance.) Taken together, these results are in disagreement with other findings (Kellett et al., 2005), which suggest that basal thermotolerance is positively related to inducible thermotolerance in Drosophila. We suggest that the physiological mechanisms supporting high basal TKD thermotolerance in our HN lines differ from those required for high inducible thermotolerance.

This work was supported by NSF DEB-0344273 to G.W.G. and D.M.R. We wish to thank Katie Provost for technical assistance and two anonymous reviewers for their insightful comments.

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-1025.
Zatsepina, O. G., Velikodvorskaia, V. V., Molodtsov, V. B.,Garbuz, D., Lerman, D. N., Bettencourt, B. R., Feder, M. E. and Evgenev, M. B. (
2001
). A Drosophila melanogasterstrain from sub-equatorial Africa has exceptional thermotolerance but decreased HSP70 expression.
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1869
-1881.


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# Calculate bond returns from yields

I have to construct and evaluate portfolio of bonds and stocks, namely I need to get return on portfolio, standard deviation and sharpe ratios. I have weekly data that contains stock prices, and I could find only one source of weekly data for the bonds. It's from Federal Reserve Economic Data, weekly Yields on Moody's AAA bonds (long-term bonds, 20-30 years to maturity) in percentages. My problem is that I have no idea how to transform weekly yields into the form comparable to stocks' weekly returns. Can anyone give me some advise on it? The data looks like this:

1. week1 7.44
2. week2 7.43
3. week3 7.40

Unfortunately I don't think it's possible to compute returns purely based on yields...

There are a few options:

1. If you're on the buy side, you can easily get access to Barclay, Citi, or BofA's bond indices. These are very high quality datasets for studying historical bond returns.

2. If you have Bloomberg, they've started providing bond indices as well. They come bundled with your Bloomberg subscription.

3. I built some bond return indices myself using the Fed's fitted yield curve. I've published the entire dataset on my blog. The advantage of this dataset is that the history is pretty long (starting in the 1960s). The disadvantage is that they outperform comparable benchmark issues in a few sub-periods. The reasons for the discrepancy are detailed on the download page.

It may not be possible to compute returns solely on yields. However, @Oleg has information on maturity (long term bonds, 20-30 years to maturity), and the YTM gives us a coupon for an "on the run" bond. As a proxy for this bond group, you could use a bond with 25 years left to maturity with an annual coupon of 7.44, where today was the coupon date, and the coupon was paid. The bond is valued at par to yield 7.44 as the YTM.

You can use whatever software you like. Using the Kona language from https://github.com/kevinlawler/kona

The stream of 25 future Cash Flows (CF) is

CF : ( 24 # 7.44 ), 107.44

The timings of the future Cash Flows (TM) is:

TM : 1 + !25

The value of the bond today is:

+/ CF % 1.0744 ^ TM

which is 100.0

The value of the bond a week from now (when the YTM has changed to 7.43) is:

+/ CF % 1.0743 ^ TM - 7. % 365.

which is 100.2499

The value of the bond 2 weeks from now (when the YTM has changed to 7.40) is:

+/ CF % 1.0740 ^ TM - 14. % 365.

which is 100.7253

The rate of return for week 1 is (100 * -1 + 100.2499 % 100.0)

which is 0.2499 percent

The rate of return for week 2 is (100 * -1 + 100.7253 % 100.2499)

which is 0.4742 percent

I got an alert from Kona that someone viewed Kona coming from this question ... so, I am taking the opportunity to add two thoughts.

Duration is the present value weighted timing of all future cash flows from a bond. So, (as @Heilin states in "his" comment) it can be used to get an APPROXIMATE return.

However, the first algorithm calculates the ACTUAL present value of the 25 year proxy bond at the "current" time and at a week later, when yields have changed. As such, the calculation takes into account both the static yield income (the nominal coupon cash flows are unchanged, but the timing is 7 days shorter), and the duration impact (all nominal flows are transformed to their present values, which are determined by the timing of each flow, and the yield at that time).

You can calculate an approximation. Yields are quoted on an annual basis. Bond valuations are based on Discounted Cash Flow formulas. Let’s take your sample data: weekly yields of 7.44, 7.43 and 7.40.

$100 invested for a year at a yield of 7.44% will be worth 107.44 at the end of a year. That is 100 x (1.0744 ^ ( 365 / 365 )) = 107.44. The rate of return for the year is 7.44% The value at the end of a week would be 100 x (1.0744 ^ ( 7 / 365 )) = 100.1377. This translates to a weekly rate of return of 0.1377% For the weekly yield of 7.43% we have 100 x (1.0743 ^ ( 7 / 365)) = 100.1375. This is a weekly rate of return of 0.1375% For the weekly yield of 7.40% we have 100 x (1.0740 ^ ( 7 / 365)) = 100.137. Similarly, that is a weekly rate of return of 0.137% • This is completely wrong... You are only accounting for the static yield income, but ignoring the duration impact. A bond's holding period return is approximately the sum of its yield income + return from changes in yield. More specifically, the approximate return over 1-week should be$\text{yield} \times 7/365 - \text{duration} \times \text{changes in yield}\$. For the same change in yield, a 30-year bond will have a vastly different return from a 2-year bond... Oct 13 '14 at 3:03
• @haginile: The answer that you initially gave was that it is not possible. Now, suddenly, we have 2 algorithms. I think that is a great improvement. Which algorithm is "better" depends on the circumstances. My algorithm assumes that the yield is constant for the week, and that the weekly investment periods are independent (a simplifying assumption, but appropriate in many circumstances). Since we do not have duration information, your algorithm is probably not usable. Oct 13 '14 at 11:18
• One possible way to improve the result would be to approximate the duration. @Oleg states that he is using Moody's AAA long-term bonds with a maturity of 20-30 years. If we make a simplifying assumption of a 25 year maturity and a par bond (coupon set so that the bond trades at par, given the yield), then we could estimate a duration. Oct 13 '14 at 11:54
• Using the duration calculator at investopedia.com/calculator/bonddurcdate.aspx and the assumptions: Par = 100, YTM = 7.44, Annual rate = 7.44, maturity = 25, and quarterly payments, we get a duration of 11.52. (Again, not a perfect answer, but the approximations are getting better.) Oct 13 '14 at 12:02

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# Euler's Derivation of the Cubic Formula

In his 1770 texbook, Elements of Algebra Euler gave a solution to the depressed equation x^3 = mx + n.

Assuming x = \sqrt[3]{p} + \sqrt[3]{q},  Euler cubed both sides:

 x^3 = p + 3\sqrt[3]{p^2q} + 3\sqrt[3]{pq^2} + q = 3\sqrt[3]{pq}(\sqrt[3]{p} + \sqrt[3]{q}) + (p + q) = 3\sqrt[3]{pq}x + (p + q).

The resulting equation obviously has the same structure as the original depressed cubic equation. This suggests letting 3\sqrt[3]{pq} = m  and p + q = n; and subsequently finding p   and q  and then also x = \sqrt[3]{p} + \sqrt[3]{q}.

So this is what Euler did. 4pq = 4m^3/27  and from p + q = n   p^2 + 2pq + q^2 = n^2. Combining these yields

(p^2 +2pq +q^2) - 4pq = n^2 - \frac{4m^3}{27}

which simplifies to

(p - q)^2 = n^2 - \frac{4m^3}{27}.

Thus

p - q = \sqrt{n^2 - \frac{4m^3}{27}}.

Adding and subtracting p + q = n,  gives

 2p = n + \sqrt{n^2 - \frac{m^3}{27}} 2q = n - \sqrt{n^2 - \frac{m^3}{27}}

So that the solution of the original cubic is

x = \sqrt[3]{\frac{n}{2} + \sqrt{\frac{n^2}{4} - \frac{m^3}{27}} } + \sqrt[3]{\frac{n}{2} - \sqrt{\frac{n^2}{4} - \frac{m^3}{27}} }.

## Reference

1. W. Dunham, Euler: The Master of Us All, MAA, 1999, pp. 83-84


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# Characteristic impedance of AWG 33 copper magnet wire

1. ### CopyOfA

24
I am working on impedance measurements of close-wound electromagnetic coils. I am using an Agilent LCR meter to measure the impedance of these coils over a frequency range of 20Hz-2MHz. When I perform the measurements, I get impedance magnitude and phase angle in degrees. I would like to construct a Smith chart of these measurements, but I am unsure of how to measure the characteristic impedance of the coils. These are single strand, AWG 33 copper wires - diameter of approximately 180μm including the insulation. Thus, they are not traditional transmission lines, based on my understanding. What is the best method to measure the characteristic impedance of these coils? Is it simply the DC resistance? Thanks for your help.

2. ### Baluncore

3,012
A coil is not really a transmission line, it has an impedance, but not a characteristic impedance. AWG 33 copper magnet wire would only have a characteristic impedance when part of a transmission line such as when in conjunction with another parallel strand or above a ground plane.

The impedance of your coils at a particular frequency will be the inductive reactance in series with the wire resistance. The resistance will be slightly dependent on frequency due to skin effect. There will also be a slight parallel capacitive reactance, (negative), due to lead and terminal capacitance.

Measure the inductance and series resistance with the meter. Compute reactance from inductance.
Normalise the values to the chart reference impedance, then plot them on the Smith Chart.

3. ### CopyOfA

24
Thanks for your help. I'm a bit of a newbie to the whole transmission line theory. I'm hoping to perform some reflectometry measurements on the coil -- do you think this is possible? Or do you have any literature that could point to the use of reflectometry on single wires (i.e. those that are not transmission lines)? I've also read that the reflection coefficient can be derived from the complex impedance. Again, it seems this requires the knowledge of some kind of characteristic impedance...

In performing these impedance measurements on the coils, I've come to some of the same realizations that you pointed out. In the lower frequency range, the coils are dominated by resistance in that the impedance Bode plot is essentially a flat line with no slope. However, as the frequency increases the slope changes to +20 dB/decade, indicating an inductive region. At some point in the response, the coil reaches anti-resonance (maximum value of the impedance), but then begins to decrease at -20 dB/decade, indicating a capacitive region. I think this capacitive region is due to the effects of the turn-to-turn insulation capacitance of the coil.

In order to compute the reactance and resistance could I not simply separate the real and imaginary portions of the complex impedance? If I do this, the resistance is highly frequency dependent.

4. ### Baluncore

3,012
No. Reflectometry requires a transmission line. Unless you do something unusual, such as wide space your coil on a conductive core, you do not actually have a transmission line. You might connect your coil as a termination at the end of a transmission line, then observe the reflection of the line to load mismatch. It won't do you much good.

The resistance is not highly frequency dependent, it will only change slightly over the frequency range. Compute the skin effect to estimate the RF resistance.

You know the inductance from your lower frequency measurements.
You know the parallel capacitance from your higher frequency measurements.
You measured series resistance over the frequency range.

For any frequency you can solve numerically for the complex Z = Parallel( XC, Series(R, XL) )

5. ### CopyOfA

24
Sorry to reply so late; I was out of the country with limited internet access. If you have some time, I have some remaining questions.

I've noticed something entirely different in my experiments. Here are some resistance and reactance measurements from a couple of coils that I've worked with.

I computed these values by splitting real and imaginary impedance measurements taken on an Agilent e4980A LCR meter.

$$Z=\left|Z\right|e^{j\theta} = R + jX$$
where, ##R = \Re\left\{\left|Z\right|e^{j\theta}\right\}## and ##X = \Im\left\{\left|Z\right|e^{j\theta}\right\}##.

Your last response seemed to imply that if I removed the skin effect from the resistance measurements, the resistance would remain essentially constant over the frequency range. Based on my understanding of the skin effect, I would expect the resistance to increase as frequency increases, but that does not happen here. Do you have any thoughts as to the reason for this? Again, thanks for your help.

6. ### Baluncore

3,012
It appears that up to 100 kHz things are well behaved.

Where the reactance changes sign there is a resonance.
There are a couple of resonances, one at about 500 kHz and another at 1 MHz.
The very high Q of the resonances shows the resistance is much lower than the reactance at resonance.

It appears the instrument is being dominated by the reactance at resonance.
A series resonant element in series with the DC resistance will look like resistance.
It is being fooled about the resistive component.
That could be because of the multiple turns and cross coupling.
The instrument can only tell the difference between R and jX by the phase of the returned signal.

I suggest that you read the resistance at 100 kHz, then replace the coil with an equivalent resistor. That will tell you if the resonance is in your instrument / leads.

The resistance should steadily rise in proportion to the square root of the frequency.
That is from the skin effect equation.
All other changes to resistance estimates are measurement artefacts.

7. ### marcusl

2,138
I wouldn't say that the Q is "very high". The width of the resonance divided by its center freq looks from the plot to be of order 5--well within expected values for a simple coil. BTW, the resonance is caused by capacitance between the turns in parallel with the inductance,

8. ### Baluncore

3,012
We are in the dark here. We do not know the size of these coils, but I would be very surprised if it was a case of adjacent turn resonance. I would expect the capacitance and inductance of two adjacent turns to be very small and so give a broad resonance at hundreds of MHz, not sharp, at 1MHz. I expect the resonance demonstrated here is due to the capacitance of the terminals and connections to the meter, reacting with the bulk inductance of the coil.

The coil is AWG33, close wound. We need to know;
What is the coil length?
What diameter?
How many turns on the coil?
How is the connection made to the meter?

3,973

Debra,

Dave

10. ### The Electrician

856
Try plotting your data with a logarithmic vertical scale.

11. ### CopyOfA

24
Thanks for all the input. I'll try to answer Baluncore's questions.

1) Wire is AWG 33, which implies a wire diameter of ~180μm. The diameter of the copper is ~173μm and there is ~3.5μm of insulation thickness surrounding the copper. The insulation is rated as Class F.

2) Coil length is unknown. However, the resistance of Coil 1 is approximately 11.4Ω and Coil 2 is 11.2Ω. Using the geometry and resistivity of copper (1.678e-8), the approximate length can be determined. I've calculated the lengths as: ~63.9m and 62.8m for Coil 1 and Coil 2, respectively.

3) Inner diameter of the bobbin is 0.568in or 1.44cm

4) How many turns is unknown. There are approximately 2 layers of coil wrapped around the central bobbin. See pictures below.

Coil1:

Coil 2:

5) Connection to the LCR meter -- see pictures below.

This image only shows the connection with Coil 1, but both were connected in the same manner. The blue wires were soldered onto the ends of the magnet wire and the other ends are attached to the LCR meter via the connection box shown. All measurements were made while the coil was placed on the solenoid valve stem as shown. Thus, there is a (soft) magnetic core inside the solenoid.

I've seen this resonance behavior before in much larger coils as well. For example, this coil (shown below) produces a similar impedance response.

The impedance Bode plot is shown below. Sorry, the frequency resolution is not as good with this one, as I was first beginning to use the LCR meter. And the magnitude units should read as dBΩ and not just Ω...

The LCR meter outputs ##\left|Z\right|## measured in Ohms and ##\theta## measured in degrees.

I've made measurements with the coil (only for Coil 1) attached directly to the LCR meter as shown below. The impedance measurements were taken with and without the (softly) magnetic core.

Measurement setup:

Impedance Bode plot with core:

Impedance Bode plot without core:

Thanks again for all the assistance. I really appreciate the input.

Last edited: Jul 23, 2014
12. ### jim hardy

5,321
Hmmm. at 100khz 50 ohms with core, 55 ohms with no core ?

But at 100 hz, maybe 26 with core, 22 without.

interesting.

13. ### Baluncore

3,012
In post #11, the "coil 2" picture shows a metal sleeve inside the bobbin. If that is stainless steel it will reduce the inductance very slightly. These coils have a ferromagnetic conductive core while being tested. Above about 10 kHz, skin effect will eliminate almost all of that ferromagnetic material from the inductance computation.

By length of the coil I meant the solenoid length, not the wire length. I was trying to compute the inductance from the solenoid dimensions.

14. ### jim hardy

5,321
yes, skin effect in the iron as opposed to the copper?

And might it be worth twisting his test leads together to reduce enclosed area?

15. ### Baluncore

3,012
A conductive material in the core prevents the magnetic field from entering that volume of core, which distorts the ideal field, reducing the coupling between turns and that in turn reduces inductance. An iron core will increase inductance significantly at lower frequencies but less as the frequency rises, a brass, aluminium or stainless steel core will actually reduce the inductance below that of an air core.

I expect twisting the leads will probably raise the external lead capacitance and so lower the resonant frequency. The reduction in area will not reduce inductance by much and so not increase the resonant frequency. It would be an interesting experiment to try.

This circuit has a capacitance involved in the resonance that has not yet been positively identified. Where the leads come from the same end of the coil, the capacitance may be “coil end to coil end” capacitance due to the overlaying of the first layer of the coil by the second layer at the common terminal end.

16. ### The Electrician

856
The capacitance involved in the resonance(s) is not in the leads or the instrument; it's the distributed capacitance of the coil itself. There is capacitance from each turn in the coil to ALL the other turns. See here for info:

http://www.g3ynh.info/zdocs/refs/Medhurst/Med35-43.pdf

I wound an inductor similar to the OP's, but with 30 gauge wire rather than 33 gauge. There are a total of 210 turns, roughly in 2 layers:

Then I generated a plot of the impedance and phase angle of the impedance over a frequency range of 20 Hz to 5 MHz. The vertical scale of the phase goes from -100° to 100° and is linear. The vertical scale for the impedance is logarithmic as shown. Phase is yellow and impedance is green:

There are several self resonances as the sweep approaches 5 MHz; this is typical for a coil of this type.

Here's another sweep, but showing AC resistance in green:

A question for the OP. What is your purpose (goals?) in all this?

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17. ### Baluncore

3,012
Your coil shown has two layers, so it is wound back on itself. The significant capacitance is not distributed amongst all turns, but between the coil ends where they lie against each other.

That paper is considering only single layer space wound coils on pre-threaded formers. Those coils are then grounded at one end, so it is not surprising that each turn has a small capacitance to one end of the coil, and all nearby grounds.

If the coils were close wound as a single layer, then the resonance would be at a higher frequency. An orderly winding would have normal adjacent turn capacitance, but with a significant degree of capacitive isolation between turns at opposite ends of the coil. That is because the turns at the ends of the coil cannot directly “see” each other electrostatically. Only adjacent turns would effect each other as electrostatic capacitors. The resonance would be that of the capacitance of two adjacent turns with the inductance of those two turns alone, but with many of those resonant elements in series.

We now see that the coils are scramble wound with no attempt made to keep the ends or layers of the coil apart. The ends of the coil can capacitively influence each other very strongly which will have a completely different self capacitance and resonance behaviour to an orderly close or space wind. It is the two layer wind that has brought the resonant frequency significantly down, by bringing the terminal ends together.

If the coil is for an iron-cored electromagnet, self resonance will matter little, especially when the magnetic circuit is closed. The forgotten art of winding coils with low self-capacitance need not be revived.

18. ### CopyOfA

24
The sleeve inside the bobbin is non-conductive. It is used to direct the magnetic field into the stem -- these coils are used to operate a solenoid valve and so directing the field into the stem (where the actuator lies) provides more efficiency.

Sorry about that... The length of the coils are: Coil 1 - 1.023 in., Coil 2 - 0.853 in.

I too believe that the capacitance is due to the turn-to-turn interaction of the voltage differences (between the turns) with the magnet wire insulation. See http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=602562. If the link doesn't work the paper is: "Self-Capacitance of Inductors" by Massarini and Kazimierczuk.

Also see post #11, picture 4. This coil was produced as a component of a Parker solenoid valve -- not hand wound. The impedance response shows a similar behavior though the two ends of the coil do not see each other electrostatically. Below is the impedance response for this coil.

Forgive me if this sounds completely ignorant, but what is AC resistance and how is it different from impedance?

19. ### Baluncore

3,012
What is it made from, iron powder or ferrite? Does it pass all the way through the coil or is it only at the ends?

Complex impedance, Z = R + j X, where R is resistance, X is the reactance.
DC resistance is R at DC = zero frequency. AC resistance is R at non-zero frequency.

What might DC reactance be, is it always zero, – infinity or maybe undefined?

20. ### The Electrician

856
Here's a sweep of a 1 meter long piece of 30 gauge wire. This image shows the reactance (imaginary part of the impedance) in green and the resistance (real part of the impedance) in yellow. The reactance increases as the frequency increases because it is due to the inductance of the wire. The resistance is another phenomenon due to losses in the copper to the flow of electricity. Those losses occur whether the current is DC or AC. Even though the reactance increases with frequency, one might think the resistance should be constant with frequency, but due to skin effect, it does increase when the frequency gets high enough. We see a slight increase in resistance as the frequency approaches 1 MHz:

This image shows the magnitude of the impedance and AC resistance. Since impedance is denoted Z = R + jX, with R being the resistance and X the reactance, the magnitude of the impedance is given by SQRT(R^2+X^2); its value can't be less than either component (R or X). Thus, the impedance curve is coincident with the resistance curve at low frequencies, whereas the reactance goes below the resistance curve:

The next two images show the same two type sweeps but of a 1 meter piece of 10 gauge wire. The skin effect comes into play at a much lower frequency, and the AC resistance of the wire at 1 MHz is about 10 times what it is at 100 Hz:

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# Vector A has a magnitude of 10 and points in the positive x-direction. Vector B has a magnitude of 15 and make an angle of 34 degrees with the positive x-axis. What is the magnitude of A - B?

Apr 24, 2016

$8.7343$ units.

#### Explanation:

$A - B = A + \left(- B\right)$

$= 10 \angle {0}^{\circ} - 15 \angle {34}^{\circ}$

$= \sqrt{{\left(10 - 15 \cos {34}^{\circ}\right)}^{2} + {\left(15 \sin {34}^{\circ}\right)}^{2}} \angle {\tan}^{- 1} \left(\frac{- 15 \sin {34}^{\circ}}{10 - 15 \cos {34}^{\circ}}\right)$

$= 8.7343 \angle {73.808}^{\circ}$.

Hence the magnitude only is $8.7343$ units.