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# Synergistic Antimicrobial Effects of Silver/Transition-metal Combinatorial Treatments

## Abstract

Due to the emergence of multi-drug resistant strains, development of novel antibiotics has become a critical issue. One promising approach is the use of transition metals, since they exhibit rapid and significant toxicity, at low concentrations, in prokaryotic cells. Nevertheless, one main drawback of transition metals is their toxicity in eukaryotic cells. Here, we show that the barriers to use them as therapeutic agents could be mitigated by combining them with silver. We demonstrate that synergism of combinatorial treatments (Silver/transition metals, including Zn, Co, Cd, Ni, and Cu) increases up to 8-fold their antimicrobial effect, when compared to their individual effects, against E. coli and B. subtilis. We find that most combinatorial treatments exhibit synergistic antimicrobial effects at low/non-toxic concentrations to human keratinocyte cells, blast and melanoma rat cell lines. Moreover, we show that silver/(Cu, Ni, and Zn) increase prokaryotic cell permeability at sub-inhibitory concentrations, demonstrating this to be a possible mechanism of the synergistic behavior. Together, these results suggest that these combinatorial treatments will play an important role in the future development of antimicrobial agents and treatments against infections. In specific, the cytotoxicity experiments show that the combinations have great potential in the treatment of topical infections.

## Introduction

In the past three decades, global usage of antibiotics has become disproportionate, and, in the majority of instances, unsuitable1. This has led to the emergence of multi-drug resistant strains. Highly resistant Gram-negative bacteria, such as P. aeruginosa, Acinetobacter, Klebsiella and Escherichia species, have become worldwide relevant pathogens2 and are part of the ESKAPE list which includes the most current deadly pathogens3. In addition, some Gram-positive bacteria, such as methicillin-resistant Staphylococcus aureus 1, 4, 5, and vancomycin-resistant Streptococcus pneumonia 4, are pathogens also very difficult to treat. The spreading of antibiotic-resistant strains embodies an issue in which international health institutions spend millions of dollars annually5. As a result, increasingly significant attention has been paid to the development of novel antibiotics; particularly metallo-pharmaceuticals, which are compounds that involve metals due to their therapeutic action, and currently belong to a class of promising antimicrobial compounds aimed to overcome resistant strains6,7,8,9,10,11,12,13.

Transition metal species, and remarkably silver compounds, lie among the most studied metallo-pharmaceuticals14, 15. Silver compounds have been used as antimicrobial agents since ancient times. It has been reported that Greek and Roman civilizations used silver as a disinfectant for water and food, as well as a wound-healing enhancer. Moreover, silver species had been widely applied in traditional Chinese medicine6,7,8,9,10, and lately, as disinfectants for water, fruits, and vegetables in several Latin American countries16. Hereafter, during the last decade, several health supplies, such as textile materials17, 18, implants, catheters, and gauze pads7, 8, have made extensive use of silver species and engineered delivery systems involving silver derivatives as antimicrobial agents6, 7, 10, 16. Recently, these antimicrobial developments have been focused to combat multi-drug resistant bacteria19. Nonetheless, silver is not the only transition metal widely used as an antimicrobial agent; zinc20, 21 and copper22 species are known to also exhibit antimicrobial effects. More interestingly, these metals are also considered as essential metals since they have been found to play key roles in many cellular processes23. However, even essential metals, when present in relatively high concentrations, become harmful24,25,26,27.

In bacteria, metal ions can inhibit growth completely at mM nominal concentrations28. In different E. coli strains this effect has been observed at >1 mM of total Co, and for sulfur-oxidizing bacteria isolated from concrete, complete growth inhibition can be observed at 5 mM Ni29. In addition, ROS stress responses have been observed at lower concentrations when using CuO nanoparticles with an E. coli biosensor at 0.1 mM Cu30. However, it is a well-documented fact that one of the main limitations in using transition metals as therapeutic agents is their toxicity to eukaryotic cells. The cytotoxicity of transition metals has been reported in the literature for human gingival fibroblast (HGF)31, TC50s of 78.7, 344, 705.8 and 872.9 µM, for Cd, Cu, Co, Ni respectively, and for HeLa cells32 the TC50 of Zn has been reported to be 600 µM.

To overcome the cytotoxicity drawbacks of using transition metals as antimicrobial agents, it has been suggested that the overall minimum inhibitory concentration can be lowered by the formulation of proper combinations of metals, resulting in effective therapies under the toxicity threshold10, 33. In this work, we hypothesized that this approach was suitable for silver/transition metal combinations, and used Escherichia coli and Bacillus subtilis as the Gram-negative and Gram-positive models, respectively, to test the synergistic effect of these combinatorial treatments. Our results support the effectiveness of this approach. When combinatorial treatments, composed of transition metals (zinc, copper, nickel, or cadmium) and silver, are tested against E. coli and B. subtilis, the combinations exhibit up to an 8-fold enhancement in their antimicrobial properties, compared to the individual treatments. We analyzed the combined antimicrobial effects and found them to be synergistic against E. coli and B. subtilis. Furthermore, we show that the minimum inhibitory concentrations of the transition metals are lowered significantly. We assessed the cytotoxicity of the treatments and found low cytotoxic effects for most combinatorial silver-transition metal treatments in three different cell lines (blasts and melanoma rat cell lines and a human keratinocyte cell line). The latter as a representation of the major cell line type found in the epidermis, which is reported to be a promising screening tool for predicting irritant and toxic skin effects34.

## Materials and Methods

### Minimum Inhibitory Concentration

Minimum Inhibitory Concentrations (MICs) were assessed in 96-well plates (Costar, Corning) based on a modified methodology reported by Andrews35 and NCCLS36. Stocks of transition metal salts, NiSO4·6H2O, CoCl2·6H2O, 3CdSO4·8H2O, CuSO4·5H2O, ZnSO4·7H2O, were prepared at final concentration of 100 mM using distilled water as the vehicle. A stock solution of 5 mM AgNO3 was also prepared under similar conditions. To perform the MIC tests, we added in the top well the necessary volume from our stocks to reach a 16 mM concentration of the transition metal within a final volume of 200 µL. Next, serial dilutions were performed by taking 100 µL from every next well into 100 µL of culture media and discarding the last 100 µL from the last dilution. This way, the tested concentrations ranged from 8 to 0.0156 mM for the transition metals after the bacteria inoculum is added. A similar procedure was performed with the silver stock, where the top well was prepared to achieve a final concentration of 3840 µM, after addition of the inoculum. The nominal concentrations of Ag ranged from 1920 to 3.75 µM.

To inoculate each test well in the MIC assay, an overnight culture (20 hours culture incubated at 37 °C–150 rpm) for each strain (Escherichia coli, ATCC-11229, or Bacillus subtilis ATCC 23857) was diluted 1:250 in fresh medium and incubated until a critical optical density (OD600 of 0.2 ± 0.02) was reached; adjusting with fresh medium if necessary in order to reach a cell concentration range between 107–108 cells/mL. Cell number was monitored by the plate-counting method measured by serial dilutions. From this, a 1:100 dilution was made with fresh medium in a 1.5 mL tube, then 100 µL of this dilution was added to each test well in order to achieve a final concentration of 105 cells/mL, following incubation at 37 °C–150 rpm.

After 20 h of incubation at 37 °C, the ODs of the control and the treated inoculum were measured. We next determined that the MIC for each of the transition metals corresponded to the concentration at which no significant growth was observed (OD600 < 0.05). All tests, and their respective control samples, were performed in replicates of 3.

### MIC determination through Checkerboard assays for the silver/transition metal combinations (STMCs)

Checkerboard assays37, 38 were performed in 96-well polystyrene plates, in order to unveil the synergistic effects of silver and transition metals. The assays tested the MIC of each transition metal and the STMCs corresponding to 0, 0.5, 0.25, and 0.125 of their respective individual MIC (Supplementary Figure S1). The MIC fractions of silver were prepared along the abscissa axis of the plate and the transition metal MIC fractions were placed along the ordinate axis of the plate. Concentrated metal solutions were prepared in culture media, so that when added to the culture the needed volume of metal and inoculum MICs fractions was reached.

Bacterial cultures were grown for 16 h at 37 °C and 150 rpm, from which a dilution (1:250) was prepared in fresh Luria Bertani broth (LB) medium, and incubated until it reached a critical optical density (OD600 = 0.20 ± 0.02), adjusting with fresh medium if necessary. This allowed us to obtain a concentration range between 107–108 cells/mL. A 1:20 dilution of this culture was adjusted in LB medium, followed by the addition of 20 µL (1:10) of the inoculum, this resulted in an estimated concentration of ~105 cells/mL within a final volume of 200 µL. The 96-well plates were incubated at 37 °C and 150 rpm for 20 h. After incubation, the ODs of the control and the treated inoculums were measured, and the respective values were recorded. Each STMC, and their corresponding control samples, were tested in triplicates per plate.

STMCs at lower concentrations were made to emulate an isobologram analysis39, 40 in 96-well plates. MIC fractions of 0.25, 0.125 and 0.062 of silver and each transition metal (M) nominal concentrations were combined to achieve final ratios of: 0.062:0, 0.062:0.25, 0.062:0.125, 0.062:0.062, 0.125:0.062, 0.25:0.062, 0:0.062 of Ag:M in a final volume of 200 µL, including the bacteria inoculum. These combinations were diluted 2-fold and 100 µL were transferred each time after a thorough mixing and discarding the last 100 µL from the 5th dilution.

The inoculation of these wells was performed using the same procedure described in the MIC section. The 96-well plates were incubated at 37 °C and 150 rpm for 20 h. After incubation, the ODs of the control and the treated inoculums were measured, and the respective values were recorded. Each STMC additional dilutions, and their corresponding control samples, were tested in triplicates.

### Antibacterial effect of STMCs

One STMC per transition metal used was selected to assess its bactericidal activity at 1 h. The antimicrobial effect was tested in an overnight culture of E. coli ATCC 11229 that was diluted (1:250) in 250 mL of LB medium, followed by incubation at 37 °C-150 rpm until it reached exponential phase at a critical optical density (OD600 = 0.10–0.20). From this medium, 25 mL were transferred to 250 mL flasks, with the addition of their corresponding amount of STMC solution, and incubated at 37 °C and 150 rpm for 1 h. This time point was chosen in order to observe if there was a short-time exposure bactericidal effect. After incubation CFU/mL were determined through the drop-count method. 10-fold diluted aliquots were obtained and, three 10 µL drops of each of the diluted samples were placed on LB agar plates. The plates were incubated at 37 °C during 24 h and at that time the viable bacterial colonies were counted. All tests, and their respective control samples, were performed in replicates of three.

### Flow cytometry analysis

STMCs were prepared as described in the checkerboard methodology. The bacteria inoculum was incubated at 37 °C and 150 rpm for 20 h. After the incubation time, the bacteria were chilled on ice until they were analyzed. Flow cytometry with a propidium iodide dye was used to detect differences in membrane permeability10. A double-stain procedure with two fluorescent nucleic acid stains was followed as previously described41. Briefly, 106 bacteria were directly labeled and incubated (15 min, dark, room temperature) with propidium iodide 4 μg (PI, Sigma Aldrich, St. Louis, MO., USA) and SyBR-G 1X (Life Technologies, Carlsbad, CA., USA) in 250 μL PBS. For calibration purposes a sample of each bacteria was heated for 10 min at 96 °C as a positive control. Analysis was performed in a BD FACS Canto II cytometer (BD, San Jose, CA., USA) configured with channel PerCP (670 LP nm band-pass filter) and FITC (530/30 nm band-pass filter) for PI and SyBR-G fluorescence acquisition, respectively. A total of 10,000 events per sample were collected at low flow rate. Signals were collected in a logarithmic scale in FSC and SSC and bacteria were double discriminated on green (live) and red (dead-permeable) fluorescence; double negative signals (debris) were also evaluated. Data were analyzed with the FlowJo software v.8.8.6 (Tree Star, Inc., Ashland, OR., USA) and the experiment was independently replicated three times.

### Cytotoxicity assay in mammalian cells

HaCaT cells, a spontaneously transformed human keratinocyte cell line, (kindly provided by Dr. Pablo Ortiz, CIBO, Mexico) were cultured in DMEM medium supplemented (sDMEM) with 1 g/L glucose, 4 mM L-alanyl-glutamine, 10% heat-inactivated fetal bovine serum (FBS; Gibco, NY, US), 50 IU/penicillin and 50 μg/ml streptomycin (Sigma, Israel) at 37 °C and 5% CO2. To measure mitochondrial activity as a viability marker, the MTT assay was performed in triplicates, as described previously42 with little modifications. To expose the cultures to the individual metals and the STMC treatments, 96-well microplates were seeded with 4 × 104 cells in 100 μl of sDMEM and left to attach overnight. Next, the old medium was replaced with 200 μl of sDMEM previously formulated with the different concentrations of transition metals. 10% DMSO was used as a positive control and medium alone as the negative control. After a 24 h exposure, the medium was discarded and 100 μl of fresh MTT solution (0.5 mg/ml of MTT in DMEM medium without FBS) was added to the cells and they were incubated for 4 h at 37 °C and 5% CO2. The MTT solution was substituted with 100 μl of Isopropanol-0.04 N HCl, in order to dissolve the formazan crystals and the optical density (OD595nm) (655 nm in the reference) was measured in an iMark microplate reader (Bio-Rad, Tokyo, Japan). Viability was calculated as the ratio of the mean between the OD595nm of the treated groups and the OD595nm of the negative control.

H9c2 cells (Rat cardiomyoblast, CRL-1446, ATCC, Manassas, VA, USA) were maintained at 37 °C in a 5% CO2 humid atmosphere in 25 cm2 culture flasks in DMEM media supplemented with 10% fetal bovine serum (FBS), penicillin (100 U/mL) and streptomycin (100 µg/mL). Cytotoxicity of transition metals was analyzed on H9c2 cells. Briefly, once H9c2 cells reached 70–80% confluence in the 25 cm2 flasks, cells were plated into 96 wells trays at a density of 2 × 103 cells per well in 100 µL medium. Mouse skin melanocytes (B16F10 cells, CRL-6475, ATCC, Manassas, VA) were maintained at the same culture conditions described above, except that they were in a DMEM/F12 culture medium supplemented with 10% FBS and 1X antibiotic-antimitotic (all from ThermoFisher Scientific, Waltham, MA) and plated into 96 wells plates at a cell density of 5 × 103 cells per well in 100 µL of medium. For these cell lines, twenty-four hours after they were plated, the different treatments were added and then the treated cells were incubated for 48 h. Cytotoxicity was assessed, for both cell lines, by the AlamarBlue® assay (Invitrogen, Eugene, OR) according to the manufacturer instructions. The half maximal inhibitory concentration (IC50) was determined using the software Origin (Northampton, MA), as previously reported43. All experiments, and their respective controls, were performed in triplicates.

### Data analysis and Calculations of Free Ions in Solution

To estimate the significance of the observed differences between the treatments employed, all collected data was subjected to an analysis of variance (ANOVA) and Fisher’s least significant difference (LSD) tests, using Microsoft Excel 2013.

The interaction of silver with transition metals was analyzed using the Bliss independence model described by Hegreness et al.4, which states that the interaction can be considered as synergistic when the combined effect of the antimicrobial agents is greater than the predicted effect of its individual components. Therefore, the S value; the difference between the predicted value of individual components x and y (f x0 and f 0y , respectively) and the combined treatment xy (f xy ) is denoted in the form:

$S= ( f x 0 f 00 ) ( f 0 y f 00 ) − f x y f 00 BlissIndependenceModel$
(1)

where: f x0 = treatment with Ag, f 0y  = treatment with the transition metal, f xy  = treatment Ag + metal, f 00 = growth control of a culture that has not been treated with the transition metals. The value of S describes the interaction between combinatorial treatments as follows: S > 0 Synergistic; S = 0 Additive; S < 0 Antagonistic.

Speciation for metals and STMCs was performed using the Visual MINTEQ 3.1 program designed and developed by Jon Petter Gustafsson at KTH in Sweden44. Metal species were calculated using medium formulations as reported by manufacturers (BD, ThermoFisher Scientific, Sigma-Aldrich) at 37 °C, and pH of 6.8 and 7.0 for LB and DMEM mediums, respectively.

## Results and Discussion

### Microbial Toxicity of Transition Metals in Prokaryotic Cells

Since the times of the first human civilizations, society has taken advantage of the antimicrobial properties exhibited by different metals22, 45, 46. Nowadays, there is a wide variety of compounds and materials that exhibit antimicrobial activity due to their transition metal content, such as Ag, Zn and Cu21. We here hypothesize that treatments involving silver salts and transition metal combinations (STMCs) will exhibit synergistic antimicrobial effects, when compared to the effect of the individual compounds. To evaluate the properties of the different STMCs, we analyzed the synergistic effects of antimicrobial activity in E. coli (Gram-negative microorganism) and B. subtilis (Gram-positive microorganism).

We first tested Zn, Cd, Ni, Co, Cu and Ag at different concentrations to determine their individual MIC values. The summary of the observed MIC values is displayed in Table 1. Most of the metals tested in E. coli, Zn, Cd, Ni, Co, and Cu, Ni, Co and those in the assays for B. subtilis have MICs between 1 and 2 mM. Only Cu, with an MIC of 4 mM in E. coli assays, and Ag with an MIC of 60 µM for both microorganisms, were found apart from the range previously mentioned. All of these MICs correlate with those found in the literature10, 47, 48. Moreover, the available free metal ions in the medium may vary due to its composition; therefore, total free ions were calculated and are reported in parenthesis in Table 1 and also in Supplementary Tables S1 and S2. As can be observed in Tables 1, S1 and S2, the percentage of the total metal available as free ions varies depending on the metal and its concentration.

### Synergistic effect of silver/transition metal STMCs

We tested combinations of Ag with different transition metals (Cu, Co, Zn, Ni and Cd) via a checkerboard assay, a widely used methodology to test synergistic effects between antimicrobial agents38, 49,50,51. The inhibitory effect of the different transition metal combinations with silver against E. coli is shown in Fig. 1. It can be observed that a reduction to 0.25 of the MIC of silver is achieved when combined with Zn, Cd, Ni, and Cu. Moreover, a reduction to 0.5 of the MIC of each of the transition metals is shown, respective to the values when tested alone (Fig. 1A–D). Cd, Ni, and Cu were able to enhance the antimicrobial activity of silver at both: 0.5 and 0.25 of their individual MICs (Fig. 1A,B,D), whereas similar behavior was observed in the case of Zn, when present at 0.5 of its MIC, albeit in a lower extent (Fig. 1C). All the STMCs, with inhibitory activities above 80%, exhibited statistically significant differences when compared to the treatments with individual transition-metal salts (p < 0.05). Only the STMCs made with Co were not able to increase the inhibitory effect of the silver MIC fractions.

Figure 2 shows that the transition metal that best enhanced the inhibitory activity of Ag against Bacillus subtilis was Zn, since it decreased its MIC to 0.125 and 0.25 when combined with 0.5 of the MIC of Ag. For the rest of the metals, when combining 0.5 MIC of each individual transition metal, the experiments show that >80% inhibition can be achieved with each of the fractional Ag MICs tested. For the case of Cu the data shows that at 0.5 and 0.25 of the Cu MIC, the MICs of Ag can be decreased four and two-fold, respectively, to achieve >80% inhibition in B. subtilis. Ni and Co at 0.5 MIC can enhance by two-fold the inhibitory effect of Ag. The calculated free ions of the STMCs tested in both bacteria are reported in Supplementary Tables S3 (E. coli) and S4 (B. subtilis).

The results described here have also been observed with different transition metals in the form of complexes against Gram-positive bacteria52 and fungi53, 54. In this work, we observed that four of the five metals used, when combined with silver, were able to increase the inhibitory effect of silver MIC fractions in E. coli. Only Co did not show this effect significantly. Similarly, in the assays made with B. subtilis, four of the five metals were able to increase the effects of Ag. Only for the case of Cd we did not observe the phenotype. Previously, different reports have used transition metals (primarily in nanoparticle or polymeric systems) in combination with silver to take advantage of its inhibitory effects; such is the case of Cu55, Zn56 and Co57.

The nature of the interactions between the components of the STMCs, in virtue of their antimicrobial effect against E. coli, were assessed as a function of their individual and combined activities using the Bliss Mathematical Model represented in Equation 1. These results are displayed in Fig. 3 and they show that the overall response appears to be synergistic. Even in the samples with inhibitory activities below 80%, the effect was slightly higher than the individual counterparts (Fig. 3A–D). In the case of Ni and Cu, an antagonistic interaction with Ag can be observed (Fig. 3A,B), however, this behavior is not present when one of the concentrations of the components increases, particularly in Ni, which exhibits synergistic effects when the concentration of Ag is found at 0.5 of its MIC (Fig. 3A).

Figure 4 shows the interactions between components in the B. subtilis assays. The interaction between Ag-Ni can be classified as synergistic, especially in the combinations of 0.125, 0.25 and 0.5 MIC of Ni with a 0.25 MIC of Ag. However, we can see that in combinations with high concentrations of Ag the interaction is antagonistic, changing with the increase of Ni. In the same manner, at low concentrations of Ag and Cu the interactions are also antagonistic, but they switch to synergistic with combinations of 0.25 and 0.5 MIC of Ag with 0.5, 0.25 and 0.125 MIC of Cu. The same synergistic effect can be observed with Ag-Zn at 0.25 and 0.5 MIC of Ag with 0.5, 0.25 and 0.125 MIC of Zn. With Co the only combinations that can be classified as synergistic are 0.5 MIC of Ag with 0.5 MIC of Co and 0.5 MIC of Ag with 0.25 and 0.5 of Co.

To explore a possible hormesis effect, exhibited in some biological systems where effects (such as growth) are accentuated in the presence of low concentrations of a compound58, we performed treatments at lower concentrations of the STMCs. The results demonstrated that no significant inhibitory or enhanced growth effects where shown for most of the STMCs. Only for the case of Ag-Ni and Ag-Co a slight hormesis effect was observed in B. subtilis. (Supplementary Tables S5 and S6).

Even though there is a wide array of literature focused on combination of antimicrobial agents and their enhanced effects as a combined treatment, the nature of the interaction of the combined components has not been widely explored. Different works have reported conditions where both synergistic and antagonist effects have been observed between Ag-Cu, the interaction depends on the bacterial species treated59. In this work, it can be observed that for the case of Gram-negative E. coli, most of the STMCs showed strong synergistic effects. In Gram-positive B. subtilis, the same combinations, although exhibited synergistic behavior, most of them are closer to an additive behavior. As can be seen in Figs 3 and 4, most of the synergistic interactions are found at combinations between the higher MIC fractions, and as the MICs fractions decrease, the interactions shift to an antagonist nature. This effect, although stronger for our experiments with Gram-positive bacteria, was also observed in our assays with Gram-negative bacteria. A very similar behavior has been previously reported for combinations with Ag-Cu tested in L. pneumophila 60. In other cases, as shown in Fig. 4, an STMC can exhibit synergistic behavior, but when the concentration of one of the components decreases the synergistic effects increases. A reason for this could be that, since some of the STMCs have similar cellular targets, when high concentrations are present one of the components could be acting in an overlapping target, but when lowering the concentration, both can exert its antimicrobial activity separately27.

### Enhanced bactericidal activity of Ag in silver/transition metal STMCs

To determine the ability of each transition metal and the STMCs to kill both Gram-negative and Gram-positive bacteria, we studied their bactericidal effects by exposing exponentially growing bacteria to different treatments. Figure 5 provides a representation of the bactericidal activity found for each STMC in E. coli. Evidently, Cd, Cu and Zn (Fig. 5B–D) were found to significantly enhance the bactericidal activity of Ag against viable E. coli in exponential phase (p < 0.5). In addition, the measured data demonstrates that the enhanced activity of silver reduces cell concentration (CFU/mL) between 2 and 4-log, compared to the decrease associated with the individual Ag treatment. However, Ag-Ni combinatorial treatments exhibited activities resembling that of the individual Ag antimicrobial effect (Fig. 5A), even though Ni was found to enhance the inhibitory activity of Ag (Fig. 2A). The same assays were performed with B. subtilis and the inhibitory STMCs; however, the results show no significant enhanced bactericidal effect (Supplementary Figure S2A–D).

### Flow cytometry analysis

The mechanism of antimicrobial synergy between silver and transition metals was explored by running cytometry assays using propidium iodide to determine changes in permeability. The Ag-Cu, Ag-Ni and Ag-Zn STMCs were selected since they showed inhibitory effects in both, Gram-positive and Gram-negative bacteria. At high Ag concentrations (30 μM) with the different transition metals at 1 mM (Cu, Ni, or Zn), cell death (Supplementary Fig. S3A,D) and lysis was clearly observed for both microorganisms. These STMCs also showed a high percentage of debris (Supplementary Fig. S3C,F) and in addition, an increased permeability was noticed (Supplementary Fig. S3B,E). Lower doses at half the lethal dose concentrations (Ag 15 μM) in combination with 0.5 mM (Cu, Ni, or Zn) show that across the range of concentrations, none or very slight cell death is observed for both E. coli (Fig. 6A) and B. subtilis (Fig. 6C). However, the permeability of the cells increases significantly between 20 to 50% for both microorganisms (Fig. 6B,D). Since bacterial membranes contain polymers with highly electronegative chemical groups, these can serve as sites for metal cation adsorption. Because of their ability to coordinate metals, it has been postulated that the membrane is the site at which some metals exert bactericidal toxicity. For Ag treatments, there are reports that show that Gram-negative and Gram-positive bacteria exhibit morphological changes, like the formation of a big cap between the cytoplasm membrane and the cell wall, causing serious damage on the latter61, as well as induction of OH· production10. Additional works have also shown that Zn, as ZnO nanoparticles, can cause disruption of membrane morphology and induce oxidative stress62 in both Gram positive and Gram negative bacteria, similar to the effects shown with Ag treatments. On the other hand, Ni, Cu and Cd have been reported to cause disruption of biological processes within the cell, causing DNA alterations63, and increased lipid peroxidation22, 64, 65. Therefore, our results show that one possible mechanism of synergy between the STMCs, in both B. subtilis and E. coli, could be an initial increase in permeability and damage of the cell wall followed by disruption of different internal cellular processes.

### Toxicity Data for the Different Transition Metals

A data analysis was performed to compare the total MICs and the effective bactericidal total concentrations of the individual transition metal treatments with those concentrations of transition metals previously reported as toxic in the literature. Table 2 shows the different toxic concentrations of transition metals and their effects reported in the literature for rats/mice and humans. As can be seen, some of the total MICs used here are within or slightly higher than the range of doses reported as lethal to rats/mice. For the case of E. coli the total concentration MICs of Co, Cd and Ni at 1 mM (59 ppm), 2 mM (225 ppm) and 2 mM (117 ppm), respectively, meet the above criteria. Meanwhile with B. subtilis only MICs of Co and Ni at 2 mM (118 ppm) and 2 mM (117 ppm), respectively, can be considered within the range of the reported non-lethal doses.

However, as previously described in this work, when testing the metals in combinations with Ag, the MICs were reduced by different fractions, depending on the metal and the microorganism. In E. coli, the total concentrations used for the combinations were Cu 1 mM (64 ppm), Zn 1 mM (65 ppm), Cd 0.5 mM (56 ppm) and Ni 0.5 mM (59 ppm). In B. subtilis these concentrations were Cu 0.5 mM (32 ppm), Zn 0.03 mM (16 ppm), Co 0.25 mM (15 ppm) and Ni 1 mM (59 ppm). Interestingly, the final concentrations used for the combinatorial treatments are all lower than those reported as lethal for mice and rats. However, these concentrations are still higher than those reported as harmful in human oral cases.

To explore the cytotoxic effects of the transition metals and STMCs tested in this work, we first performed viability experiments in human keratinocytes (HaCat cells). Supplementary Figure S4 shows the viability for HaCat cells in culture treated with the transition metals used in this work, at the concentrations reported as the E. coli nominal metal MICs. Across the range of concentrations we observed no significant viability (p < 0.05). Most of the metals exhibited an average toxicity of 90%, except Ni with 55% at 24 h of treatment.

Toxicity experiments were performed in another cell line, H9c2 cells, to explore an additional toxicity assay. Figure 7A shows a calculated IC50 value of 3.12 µM for the treatment with individual Ag. This result indicates a strong cytotoxicity of the individual metal. Additional toxicity assays were performed, in H9c2 cells, for the rest of the transition metals, at the nominal concentrations at which they showed antibacterial activity (1 mM). As can be observed in Fig. 7B, Ni and Co did not produce important effects, showing only 15% and 38% cytotoxicity values, respectively. However, Cu and Zn showed about 60% cytotoxicity at the same concentration. Cd, on the other hand, showed to be moderately toxic with about 50% cytotoxicity. Additional concentrations were tested for each of the metals and, with exception of Cd, a typical dose-response effect was observed (Fig. 7C). IC50s were calculated and the results show that for all the metals, except Zn and Cd, values are higher (Table 2) than the respective MIC values reported in this work for the transition metals (Table 1). Ni showed to be the less toxic for H9c2 cells, as it needed about 16.5-fold concentration increase in order to reach its IC50. Co reaches its IC50 with a 20-fold concentration increase. Zn IC50 is similar to its MIC value. Due to the similar cytotoxicity of Cd at all tested concentrations, it was not possible to adjust a typical curve to calculate its IC50 value, but according to the trend observed, it is lower than 0.1 mM.

Even though the individual metals and STMCs showed cytotoxicity against some of the different cell lines, one explanation could be the difference in the concentrations of free ions available in the medium used in the experiments to calculate MICs (LB), compared to the concentration of free ions in the medium used to calculate the IC50 cytotoxic dose (sDMEM and DMEM). Therefore, the free ion concentrations needed to be correlated in both experiments to correctly determine the effect of the different metals and the STMCs. Supplementary Table S7 shows the calculated free ion concentrations corresponding to the nominal concentrations for each of the different media used. As can be observed, some metals, at equivalent nominal concentrations, show differences in the calculated free ion concentrations for the different culture medium. Based on this data, the STMCs for three transition metals were adjusted to perform additional cytotoxicity assays at equivalent free ion concentrations. Table 3 shows the new nominal concentrations used. A comparative of the free ions calculated for these mediums is reported in Supplementary Table S7.

Experiments on HaCat cells involving 24 h treatments with transitions metals at the equivalent free metal ions concentrations were performed. It was observed that the percentage of viability significantly increased above 90% for all of the metals (p < 0.01) except for Co at 750 μM where the toxicity was maintained (44.00 ± 1.24%) (Fig. 8). The same phenotype was noticed for the STMCs, all of them showed viabilities above 90%, with the exception of the Ag-Co STMC, with 750 μM Co, which showed a viability of 60%, and the Ag-Cu STMC at the highest concentration of 300 μM Cu, which showed a viability of 50% (Fig. 9). A positive cytotoxicity control treatment with 10% DMSO was used in both experiments, and a viability of 16.32 ± 3.54% was obtained.

As a complementary test to the MTT assay, the cells were evaluated qualitatively with an optical microscope (200X) (Figures S5 and S6). Compared to the control group, notable changes in the monolayer confluence and cell morphology, as well as an increase in the number of floating cells, were markedly seen in groups such as Co 750 μM (Fig. S5F) and also in the positive control (10% DMSO) (Fig. S5B). These results correlate well with the MTT results. On the other hand, groups such as Ag 15μM-Co 750 μM (Fig. S6C) showed a lesser change in monolayer confluence and cell morphology compared to the Co 750 μM treated group (Fig. S5F), but a considerable number of floating cells were observed as compared to control. This was also noted in the MTT assay, where viability of these groups decreased but it was not statistically significant. The rest of the treatments did not seem to suffer changes in the features assessed optically. In an interesting way, some of the groups that showed a higher viability than the control (p < 0.001) in the MTT assay, looked optically similar to the control group. Therefore, such increment can be related more to a higher metabolic activity than to a higher number of cells.

As well as with HaCat cells, STMCs at the adjusted free metal ion concentrations were performed in both H9c2 and B16F10 cells. Ag cytotoxicity on H9c2 cells is significantly inhibited when used in combination with Cu (Fig. 10B). When Ag concentration is 15 µM the cytotoxicity is about 50%, but when 80 µM of Cu is added, the cytotoxicity is reduced to about 17%. Combination of Co or Zn does not block Ag cytotoxicity on these cells (Fig. 10B). Cytotoxic effects were also analyzed on melanocytes (B16F10 cells), to observe the effect at the same total concentrations as the experiments with HaCat and H9c2 cells. The results obtained can be compared to those observed in the highly sensitive H9c2 cell line. Ag alone was not found to be cytotoxic to melanocytes (Fig. 7A). In addition, Ag/Zn or Zn alone had not significant cytotoxic effects on B16F10 cells and the combinations of the rest of the metals with Ag did not potentiate their toxicity (Fig. 10A). Taken together, these results suggest that one of the most promising combination is that of Ag with Cu since it shows potential to treat bacterial infections with limited cytotoxic effects on mammalian cells.

The differences in the Ag and STMCs cytotoxic concentrations observed between the different cell lines tested in this work could be explained by the nature of the cells. H9c2 cells are blasts, which normally are sensitive to a variety of insults and they are programmed to die to avoid passing mutations or damage to the progeny. On the other hand, B16F10 and HaCaT cells are skin melanoma and keratinocytes respectively. These cells are more resistant to different conditions as they have mutations in genes that control cell cycle, DNA repair, and apoptosis pathways72.

The MIC values obtained for each metal on both Gram-negative and Gram-positive microorganisms and the IC50s obtained in this work, although promising for the development of future therapeutics, cannot be directly compared to their reported LD50 values, as our work was made in vitro while LD50s are calculated from in vivo experiments. Even though these in vitro studies set the basis for pharmacological applications, as it has been shown in several reports73, 74, both bioavailability and biodistribution of the transition metals used here need to be obtained in animal models in order to determine their safety.

Although these concentrations were found to be higher than reported as cytotoxic in human cells and harmful in humans cases by oral administration, an additional important application of these treatments can be found in topical treatment, as seen in Table 2. Skin and soft tissue infections have a great incidence in communities and hospitals75, 76 and about 30% of diabetic patients have cutaneous disorders77. These kind of infections, like cellulitis, abscesses, diabetic foot infections and surgical site infections caused by multidrug-resistant pathogens, recently have become more common and alternatives to the current treatments are needed78,79,80. Therefore, here we open an area of opportunity for the use of transition metals in dermal treatments. The use of copper as a potential therapeutic in dermal wound healing was reported by Sen et al.81 noting copper sensitive pathways that regulate wound healing mediators. Some other authors reported the use of transition metals, especially silver, as antimicrobial agents in wound dressing materials and regeneration materials82,83,84. The STMCs reported here can be considered as potential antimicrobial agents in the manufacture of these kind of materials, since we show STMCs to be capable of inhibiting bacterial growth at lower concentrations than each metal as a separate treatment. Furthermore, as shown in this work, the use of STMCs has the advantage of achieving a synergistic effect, allowing the use of lower transition metal doses, and delaying the emergence of resistance to either85. Thus, these combinations could be used as a platform to develop novel combinatorial treatments against antibiotic resistance strains. Due to the increments in permeability shown in this work the combinations of metals and silver have potential to be combined with commonly used antibiotics to re-sensitize resistant bacteria. These would allow the decrease in doses of both, antibiotics and transition metals, and at the same time mitigate the possibility of a toxic effect on human cells.

## Conclusions

In correlation with the results provided in this work, we conclude that transition metals, Zn, Cd, Ni and Cu, are able to enhance the inhibitory and bactericidal activities of silver salts against Escherichia coli and B. subtilis; under the conditions implemented in this study. Nevertheless, the extent to which each STMC affected the behavior of Ag differed significantly. Ag/Ni STMC showed significant potentiation of bactericidal activity when compared to Ag in E. coli assays, however no significant effects were observed for treatments in B. subtilis. Cu, Ni, and Cd were able to enhance Ag inhibitory activity at 0.25 and 0.5 of their respective MICs, whereas Zn exhibited similar behavior only at 0.5 of its MIC. Meanwhile, Cu, Zn and Co were able to enhance Ag inhibitory activity at these fractions of their respective MICs in B. subtilis assays. We demonstrate that STMCs Ag-Cu, Ag-Ni and Ag-Zn exhibit their antimicrobial effect through an increase in cell permeability. The findings described herein provide significant advances in understanding the interactions and effects of combined transition metals and set the basics for future design of metallo-pharmaceutical treatments. In addition, most STMCs tested in this work exhibit antibacterial effects at non-toxic concentrations for human keratinocyte cells. Based on this data, we can conclude that dermic therapeutic applications could be possible, although further investigation need to be taken regarding non-cytotoxic concentrations. Taken together, our results suggest that due to the inherent complexity related to the synergistic effects observed in the systems studied, this work provides a platform for further investigations focused on unveiling deeper correlations to support the use of metallo-pharmaceuticals in future design of therapeutic strategies.

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## Acknowledgements

The authors would like to thank the 2012–2015 and 2015–2017 Paicyt research program, from Universidad Autónoma de Nuevo León. In addition, the CONACyT Basic Science Funding Grant No. 221332 also supported this work.

## Author information

### Affiliations

1. #### Universidad Autónoma de Nuevo León, UANL, Facultad de Ciencias Químicas, Av. Universidad s/n, Cd, Universitaria, 66451, San Nicolás de los Garza, NL, Mexico

• Javier A. Garza-Cervantes
• , Margarita Ortiz-Martínez
• , Sara Leticia Gómez-Flores
• , Jorge A. Peña-Martínez
• , Xristo Zarate
• , María Elena Cantú-Cárdenas
• , Carlos Enrique Escarcega-Gonzalez
•  & J. Rubén Morones-Ramírez
2. #### Centro de Investigacion en Biotecnologia y Nanotoxicologia, Facultad de Ciencias Quimicas, Universidad Autonoma de Nuevo Leon, Parque de Investigacion e Innovacion Tecnologica, Km. 10 autopista al Aeropuerto Internacional Mariano Escobedo, Apodaca, Nuevo Leon, 66629, Mexico

• Javier A. Garza-Cervantes
• , Xristo Zarate
• , María Elena Cantú-Cárdenas
• , Carlos Enrique Escarcega-Gonzalez
•  & J. Rubén Morones-Ramírez
3. #### Centro de Investigación y de Estudios Avanzados del IPN, Unidad Monterrey, Parque PIIT, 66600, Apodaca, Nuevo León, Mexico

• Arturo Chávez-Reyes
4. #### Cátedra de Cardiología y Medicina Vascular, Escuela de Medicina, Tecnologico de Monterrey, Monterrey, Nuevo León, Mexico

• Elena C. Castillo
•  & Gerardo García-Rivas
5. #### Centro de Investigación Biomédica, Hospital Zambrano-Hellion, Tecnologico de Monterrey, San Pedro Garza-García, Nuevo León, Mexico

• Elena C. Castillo
•  & Gerardo García-Rivas
6. #### Departamento de Microbiología, Centro de Ciencias Básicas, Universidad Autónoma de Aguascalientes, Av. Universidad 940, Colonia Ciudad Universitaria C.P., 20131, Aguascalientes, Aguascalientes, Mexico

• Oscar Antonio Ortega-Rivera
•  & Eva Salinas
7. #### Universidad Autónoma de Nuevo León, UANL, Facultad de Ciencias Biológicas, Av. Universidad s/n, Cd. Universitaria, 66451, San Nicolás de los Garza, NL, Mexico

• Alan Pepi-Molina
8. #### Universidad Autónoma de Nuevo León, UANL, Facultad de Ingeniería Mecánica y Eléctrica, Av. Universidad s/n, Cd. Universitaria, 66451, San Nicolás de los Garza, NL, Mexico

• Mario T. Treviño-González

### Contributions

J.A.G.C., A.C.R., E.C.C., G.G.R., O.A.O.R., E.S., S.L.G.F., J.A.P.M., A.P.M. and J.R.M.R. designed and performed the experimental data and wrote the manuscript. M.T.T.G. helped in the design of some experimental data and helped with the format of the manuscript. X.Z., M.E.C.C. and C.E.E.G. helped with the format and writing of the manuscript. M.O.M. helped in the initial stage of the project determining the different MICs of a range of individual transition metals.

### Competing Interests

The authors declare that they have no competing interests.

### Corresponding author

Correspondence to J. Rubén Morones-Ramírez.


text:
# Portable and efficient Caesar cipher

Implementation of a Caesar cipher is a popular exercise and there are many implementations posted here on Code Review.

My version is intended to be efficient and portable (subject to some limitations, below).

#include <algorithm>
#include <array>
#include <cctype>
#include <climits>
#include <cstdlib>
#include <iostream>
#include <iterator>
#include <numeric>
#include <stdexcept>

class caesar_rotator {
using char_table = std::array<char, UCHAR_MAX+1>;
const char_table table;

public:
caesar_rotator(int rotation) noexcept
: table{create_table(rotation)}
{}

char operator()(char c) const noexcept
{
return table[static_cast<unsigned char>(c)];
};

private:
#define LETTERS "abcdefghijklmnopqrstuvwxyz"
static char_table create_table(int rotation)
{
constexpr int len = (sizeof LETTERS) - 1; // don't count the terminating null
static const auto* alpha2 = reinterpret_cast<const unsigned char*>(LETTERS LETTERS);

// normalise to the smallest positive equivalent
rotation = (rotation % len + len) % len;

char_table table;
// begin with a identity mapping
std::iota(table.begin(), table.end(), 0);
// change the mapping of letters
for (auto i = 0;  i < len;  ++i) {
table[alpha2[i]] = alpha2[i+rotation];
table[std::toupper(alpha2[i])] = static_cast<char>(std::toupper(alpha2[i+rotation]));
}
return table;
}
#undef LETTERS
};

int main(int argc, char **argv)
{
constexpr int default_rotation = 13;
// Parse arguments
int rotation;
if (argc <= 1) {
rotation = default_rotation;
} else if (argc == 2) {
try {
std::size_t end;
rotation = std::stoi(argv[1], &end);
if (argv[1][end]) { throw std::invalid_argument(""); }
} catch (...) {
std::cerr << "Invalid Caesar shift value: " << argv[1] << " (integer required)\n";
return EXIT_FAILURE;
}
} else {
std::cerr << "Usage: " << argv[0]
<< " [NUMBER]\nCaesar-shift letters in standard input by NUMBER places (default "
<< default_rotation <<")\n";
return EXIT_FAILURE;
}

// Now filter the input
std::transform(std::istreambuf_iterator<char>{std::cin},
std::istreambuf_iterator<char>{},
std::ostreambuf_iterator<char>{std::cout},
caesar_rotator{rotation});
}

# Features and limitations

• It's written using portable C++ which compiles with any compiler supporting C++11 or later. (But don't let that inhibit reviews from suggesting improvements that require newer versions - I am still interested!)
• We can shift by any representable integer (negative shift decodes a positive shift, and vice versa).
• It works with all single-byte character sets. Unlike many implementations, it works on systems where alphabetic characters are discontinuous, such as EBCDIC.
• It works with multi-byte character sets where the alphabetic characters are invariant (e.g. UTF-8). When using codings that shift out the alpha-chars, then non-alphabetic characters may be modified - but using the same program for the decoding will correctly restore the original input text.
• Because we use a table to produce output, we need storage for that table, sufficient to represent all character values. No problem for systems where CHAR_BIT is 8, but it may prevent use on systems with much wider character type.
• There is an overhead to constructing the table each run, which won't pay for itself with short input streams. But it's still small compared with the overhead of creating a process, and it's a big win for large inputs.
• I may write a separate wide-char implementation that addresses the limitations.

# The ugly stuff

• I don't have access to an EBCDIC system to confirm my claim that it's correct on such platforms.
• Most of the ugliness is in create_table(), which in an earlier version had been an immediately-invoked lambda expression in the constructor's initialiser list; I moved it out to a named private function to make it easier to read and easier to skip when perusing the public interface.
• I don't like using a macro to concatenate two copies of the alphabet. But I couldn't find an alternative constexpr way to do that, even in C++20, other than writing the letters twice, which I didn't want to do. (And alpha2 still isn't constexpr, because of the cast, but at least it enables reasonable compilers to store the string literal in .text segment or equivalent.)
• The reinterpret_cast offends me. And the static_casts, though to a lesser extent. But we need unsigned values for std::toupper() and for array indexing, and plain char for the stream handling and string literal.
• I don't much like using % twice to get the positive modulo; is there a simpler way? I considered rotation %= len; if (rotation < 0) rotation += len; but I'm not convinced that's better.
• In the main function, I throw an exception when argument has anything following an initial number, so that I handle all parsing errors in one place. Is that okay, or is there something better I could do?
• An answer that improves any one of the "ugly" things is very welcome (even if it's just confirmation that the code functions correctly using EBCDIC). Oct 30 '21 at 12:24

# Avoid macros

There is absolutely no need for a macro here. Just write:

static const char *letters = "abcdefghijklmnopqrstuvwxyz";
auto alpha = reinterpret_cast<const unsigned char *>(letters);

And to access it safely:

table[alpha2[i]] = alpha2[(i + rotation) % len];

Alternatively, you could create a mutable string array holding the letters, and then use std::rotate() to create a rotated version of it.

# Avoiding the reinterpret_cast

Instead of creating a reinterpret-casted pointer to the string holding the letters, you can static_cast the individual characters. That's safer but arguably uglier because you then need more casts.

A much better way would be to make char_table a class with an operator[] that takes a char as an argument, and which returns a reference. That way you can write:

char_table table;
...
for (auto i = 0; i < len; i++) {
table[letters[i]] = letters[(i + rotation) % len];
...
}

Of course, that pesky std::toupper() still gives trouble. Maybe you can create a helper function that takes care of doing toupper() safely on chars.

# Optimizing table construction

Most of the inefficiency is in the modulo operation. But it can be largely avoided: first ensure rot is between 0 and 26, then you can make use of the fact that i + rot is between 0 and 52, and use i + rot < 26 ? i + rot : i + rot - 26 as an index. Again, you could make a little helper function for this to make the code more readable.

Apart from that, consider making build_table() a constexpr function. This is rather trivial with C++17 and later (the only thing to worry about is std::iota(), which isn't constexpr before C++20). It might be possible to make it constexpr in earlier C++ versions as well. With this, if rot is a compile-time constant, the table can also be built at compile-time.

If you can build one table at compile-time, you could even build all 26 of them at compile-time, making it cheap to do the Caesar substitution with a rotation that you only know at runtime.

Based on suggestions from G. Sliepen's answer, I was able to eliminate the macro and localise most of the casts:

private:
static constexpr int upper_int(char c)
{
return std::toupper(static_cast<unsigned char>(c));
}
static constexpr char upper_char(char c)
{
return static_cast<char>(upper_int(c));
}

static char_table create_table(int rotation)
{
constexpr auto* letters = "abcdefghijklmnopqrstuvwxyz";
constexpr int len = std::strlen(letters);

// normalise to the smallest positive equivalent
rotation = (rotation % len + len) % len;

char_table table;
// begin with a identity mapping
std::iota(table.begin(), table.end(), 0);
// change the mapping of letters
for (auto i = 0, target = rotation;  i < len;  ++i, ++target) {
if (target == len) {
target = 0;
}
table[letters[i]] = letters[target];
table[upper_int(letters[i])] = upper_char(letters[target]);
}
return table;
}

Not only that, but these changes reduce the maximum line length, for easier reading.
:-)

Compared to the answer that inspired this, I did the computation of target differently: here, I help branch prediction by taking the forward path all but once. Yes, that is unnecessary micro-optimisation, but I'm here for a learning experience.

For C++20 onwards, create_table() can be declared constexpr, though there's little point, as we never call it with a compile-time constant argument. With GCC, this even works with C++17, before std::iota() became constexpr - I think it's smart enough to examine the resultant code after expanding the template.

• That looks great! Oct 30 '21 at 19:18

You can avoid the reinterpret_cast without changing the logic, and at the same time drastically limit the scope of your macro:

 #define LETTERS "abcdefghijklmnopqrstuvwxyz"
+        static constexpr auto alphabet_size = sizeof LETTERS - 1;
+        static constexpr unsigned char alpha2[sizeof LETTERS * 2 - 1] = LETTERS LETTERS;
+#undef LETTERS
+
static char_table create_table(int rotation)
{
-        constexpr int len = (sizeof LETTERS) - 1; // don't count the terminating null
-        static const auto* alpha2 = reinterpret_cast<const unsigned char*>(LETTERS LETTERS);
+        constexpr int len = alphabet_size;

@@ -44,3 +47,2 @@
}
-#undef LETTERS
};

Still far from ideal (especially since it can’t be combined with AA style, which I’d otherwise follow) but much better.

Secondly, and this is certainly subjective, but I really dislike implicit conversions to bool. I just about accept omitting != nullptr; but implicit int to bool conversions just make the code less readable. Case in point, I had to actively think which exact condition was tested by if (argv[1][end]). I find it vastly more readable to write this explicitly, i.e.:

if (argv[1][end] != '\0')

I’d probably also give argv[1] a dedicated name in this scenario since it’s used repeatedly, and requires nested array access. And I’d probably make rotation const, and use a conditional expression with a nested IIFE to initialise it. However, this will require a function-level try block (and admittedly this makes the code longer, which might not be a good trade-off in this simple case).

Lastly, we probably don’t want to just catch ... because if there’s an unexpected exception that’s a bug in the code, and catching the exception obliterates the stack trace.

• I tried initializing a std::array<unsigned char> from string literal, but that's an error. I didn't realise I could initialise a unsigned char[] without a warning. I guess the array size should be alphabet_size * 2 or sizeof LETTERS * 2 - 2 to not have a couple of wasted elements? Remember that sizeof measures the whole string including the final null character that we don't need here. Oct 31 '21 at 15:59
• I did originally "scope" LETTERS around the two lines within the function, but that was hard to read. Your suggestion of extracting the constants is much better. (I only changed to a class quite late - originally, rotate() just returned a lambda containing the translation table). Oct 31 '21 at 16:01
• @TobySpeight Yes, of course, my mistake. It does need to be sizeof LETTERS - 1 though, otherwise the array will be too small to hold the string literal incl. zero termination. Oct 31 '21 at 16:01
• @TobySpeight That’s only valid in C, not in C++! Oct 31 '21 at 16:08
• Ah, dcl.init.string, and the consensus interpretation is that you're right: What language standards allow ignoring null terminators on fixed size arrays? Oct 31 '21 at 16:19

I suggest a different approach:

Nail the locale to the C locale, write the 26 possible translation-table as read-only data, and let the constructor just choose the right one.

It probably takes a bit more space in the executable (6.5K data for 8bit char), but it will certainly be faster and needs less per-instance space at runtime.

#include <array>
#include <numeric>

constexpr auto translation_tables = []{
const char* bands[]{
"abcdefghijklmnopqrstuvwxyz",
"ABCDEFGHIJKLMNOPQRSTUVWXYZ"
};
std::array<std::array<unsigned char, 1 + (unsigned char)-1>, 26> r;
for (int rotate = 0; rotate < 26; ++rotate) {
auto& table = r[rotate];
std::iota(table.begin(), table.end(), (unsigned char)'\0');
for (auto band : bands)
for (int i = 0; i < 26; ++i)
table[band[i]] = table[band[(i + rotate) % 26]];
}
return r;
}();

constexpr auto caesar_rotator(int shift) noexcept {
if ((shift %= 26) < 0)
shift += 26;
auto table = translation_tables[shift].data();
return [=](unsigned char c) noexcept { return table[c]; };
}

• You only need two consecutive copies of 'a'..'Z', not 26. Just point a pointer to the starting point based on the shift. That is, work the same as the old "decoding rings". Having the list repeat prevents the modulo operation, which is your real performance point. Nov 1 '21 at 16:41
• @JDługosz You know the two shifted rings are embedded in a block of 256 symbols (for 8-bit, aka minimal, bytes), the rest being identity-mapped? Thus, really need one mapping per shift-value, of which there are 26 different ones, as the rings are 26 unique symbols long. Nov 1 '21 at 17:23
• I see. I thought you were just avoiding the expensive modulo operation, not the range test for ischar. Nov 1 '21 at 22:33

format: plain


text:
Let $$u$$ and $$v$$ be integers satisfying $$0 < v < u$$. Let $$A = (u,v)$$, let $$B$$ be the reflection of $$A$$ across the line $$y = x$$, let $$C$$ be the reflection of $$B$$ across the y-axis, let $$D$$ be the reflection of $$C$$ across the x-axis, and let $$E$$ be the reflection of $$D$$ across the y-axis. The area of pentagon $$ABCDE$$ is $$451$$. Find $$u + v$$.

(第十八届AIME1 2000 第2题)


text:
ChiantiPy Core Classes¶

ChiantiPy.core Package¶

chianti.core - contains the main classes for ChiantiPy users.

 bunch(temperature, eDensity, wvlRange[, …]) Calculate the emission line spectrum as a function of temperature and density. continuum(ion_string, temperature[, …]) The top level class for continuum calculations. ion(ionStr[, temperature, eDensity, …]) The top level class for performing spectral calculations for an ion in the CHIANTI database. ioneq(el_or_z) Calculation ion fractions as a function of temperature assuming ionization equilibrium. ipymspectrum(temperature, eDensity, wavelength) this is the multiprocessing version of spectrum for using inside an IPython Qtconsole or notebook. mspectrum(temperature, eDensity, wavelength) this is the multiprocessing version of spectrum radLoss(temperature, eDensity[, …]) Calculate the emission spectrum as a function of temperature and density. spectrum(temperature, eDensity, wavelength) Calculate the emission spectrum as a function of temperature and density.


text:
## Chemistry 12th Edition

We know that the strongest acid that can exist in water solution is $H_3O^+$ while the strongest base that can exist in water is $OH^-.$


text:
# How to write nonanticipativity constraints?

In Multi-stage stochastic programming, we write the constraints that for scenarios $$s$$ and $$s^{\prime}$$ which have the same trajectory up to time $$t$$, should take the same value. That is, $$x_{t,s} = x_{t,s^{\prime}} \quad \quad \forall t, s,s^{\prime} \in S: \: \: (\xi_{1}^s,\dots,\xi_{t-1}^s)=(\xi_{1}^{s^{\prime}},\dots,\xi_{t-1}^{s^{\prime}})$$

I thought that for all decision variables except the first stage. For example, if $$y_{ts},z_{ts}$$ are used in different stages, I should write nonanticipativity constraints for all of them. However, in some papers, nonanticipativity constraints are used for just a subset of variables, like $$x_{ts}$$. Could you please let me know how we should decide to include a variable in nonanticipativity constraints or not.

• Is there anybody who can help me with this question? – Katatonia Mar 29 at 4:16


text:
A python metaheuristic optimization library. Currently supports Genetic Algorithms, Gravitational Search, and Cross Entropy.

## Optimal (beta)

A python metaheuristic optimization library. Built for easy extension and usage.

Warning: Optimal is in beta. API may change. I will do my best to note any breaking changes in this readme, but no guarantee is given.

Supported metaheuristics:

• Genetic algorithms (GA)

• Gravitational search algorithm (GSA)

• Cross entropy (CE)

## Installation

pip install optimal

## Usage

import math

from optimal import GenAlg
from optimal import Problem
from optimal import helpers

# The genetic algorithm uses binary solutions.
# A decode function is useful for converting the binary solution to real numbers
def decode_ackley(binary):
# Helpful functions from helpers are used to convert binary to float
# x1 and x2 range from -5.0 to 5.0
x1 = helpers.binary_to_float(binary[0:16], -5.0, 5.0)
x2 = helpers.binary_to_float(binary[16:32], -5.0, 5.0)
return x1, x2

# ackley is our fitness function
# This is how a user defines the goal of their problem
def ackley_fitness(solution):
x1, x2 = solution

# Ackley's function
# A common mathematical optimization problem
output = -20 * math.exp(-0.2 * math.sqrt(0.5 * (x1**2 + x2**2))) - math.exp(
0.5 * (math.cos(2 * math.pi * x1) + math.cos(2 * math.pi * x2))) + 20 + math.e

# You can prematurely stop the metaheuristic by returning True
# as the second return value
# Here, we consider the problem solved if the output is <= 0.01
finished = output <= 0.01

# Because this function is trying to minimize the output,
# a smaller output has a greater fitness
fitness = 1 / output

# First return argument must be a real number
# The higher the number, the better the solution
# Second return argument is a boolean, and optional
return fitness, finished

# Define a problem instance to optimize
# We can optionally include a decode function
# The optimizer will pass the decoded solution into your fitness function
# Additional fitness function and decode function parameters can also be added
ackley = Problem(ackley_fitness, decode_function=decode_ackley)

# Create a genetic algorithm with a chromosome size of 32,
# and use it to solve our problem
my_genalg = GenAlg(32)
best_solution = my_genalg.optimize(ackley)

print best_solution

Important notes:

• Fitness function must take solution as its first argument

• Fitness function must return a real number as its first return value

For further usage details, see comprehensive doc strings.

## Major Changes

### 08/27/2017

Moved a number of options from Optimizer to Optimizer.optimize

### 07/26/2017

Renamed common.random_solution_binary to common.random_binary_solution, and common.random_solution_real to common.random_real_solution

### 11/10/2016

problem now an argument of Optimizer.optimize, instead of Optimizer.__init__.

### 11/10/2016

max_iterations now an argument of Optimizer.optimize, instead of Optimizer.__init__.

### 11/8/2016

Optimizer now takes a problem instance, instead of a fitness function and kwargs.

### 11/5/2016

Library reorganized with greater reliance on __init__.py.

Optimizers can now be imported with:

from optimal import GenAlg, GSA, CrossEntropy

Etc.

## Project details

Uploaded source


text:
## Some slices.

As followup to:

here is a bit more experimentation with Paraview slice filtering. This time I saved all the point data with the escape time counts, and rendered it with a few different contours

The default slice filter places the plane in the x-y orientation:

but we can also tilt it in the Paraview render UI

and suppress the contour view to see just the slice

As a very GUI challenged user, I don’t find the interface particularly intuitive, but have at least figured out this one particular slicing task, which is kind of cool.  It’s impressive that the UI can drive interesting computational tasks without having to regenerate or reload any of the raw data itself.  This time I was using the MacOSX Paraview client, which is nicer looking than the Windows version, but has some weird glitches in the file dialogues.

## Analysis.

The graphing play above shows some apparent rotational symmetry our vector equivalent to the Mandelbrot equation

\Bx \rightarrow \Bx \Be_1 \Bx + \Bc.

It was not clear to me if this symmetry existed, as there were artifacts in the plots that made it appear that there was irregularity. However, some thought shows that this irregularity is strictly due to sampling error, and perhaps also due to limitations in the plotting software, as such an uneven surface is probably tricky to deal with.

To see this, here are the first few iterations of the Mandlebrot sequence for an arbitary starting vector $$\Bc$$.

\begin{aligned}
\Bx_0 &= \Bc \\
\Bx_1 &= \Bc \Be_1 \Bc + \Bc \\
\Bx_2 &= \lr{ \Bc \Be_1 \Bc + \Bc } \Be_1 \lr{ \Bc \Be_1 \Bc + \Bc } + \Bc \Be_1 \Bc + \Bc.
\end{aligned}

Now, what happens when we rotate the starting vector $$\Bc$$ in the $$y-z$$ plane. The rotor for such a rotation is

R = \exp\lr{ e_{23} \theta/2 },

where

\Bc \rightarrow R \Bc \tilde{R}.

Observe that if $$\Bc$$ is parallel to the x-axis, then this rotation leaves the starting point invariant, as $$\Be_1$$ commutes with $$R$$. That is

R \Be_1 \tilde{R} =
\Be_1 R \tilde{R} = \Be_1.

Let $$\Bc’ = R \Bc \tilde{R}$$, so that

\Bx_0′ = R \Bc \tilde{R} = R \Bx_0 \tilde{R} .

\begin{aligned}
\Bx_1′
&= R \Bc \tilde{R} \Be_1 R \Bc \tilde{R} + R \Bc \tilde{R} \\
&= R \Bc \Be_1 \Bc \tilde{R} + R \Bc \tilde{R} \\
&= R \lr{ \Bc \Be_1 \Bc R + \Bc } \tilde{R} \\
&= R \Bx_1 \tilde{R}.
\end{aligned}

\label{eqn:m2:n}
\begin{aligned}
\Bx_2′
&= \Bx_1′ \Be_1 \Bx_1′ + \Bc’ \\
&= R \Bx_1 \tilde{R} \Be_1 R \Bx_1 \tilde{R} + R \Bc \tilde{R} \\
&= R \Bx_1 \Be_1 \Bx_1 \tilde{R} + R \Bc \tilde{R} \\
&= R \lr{ \Bx_1 \Be_1 \Bx_1 + \Bc } \tilde{R} \\
&= R \Bx_2 \tilde{R}.
\end{aligned}

The pattern is clear. If we rotate the starting point in the y-z plane, iterating the Mandelbrot sequence results in precisely the same rotation of the x-y plane Mandelbrot sequence. So the apparent rotational symmetry in the 3D iteration of the Mandelbrot vector equation is exactly that. This is an unfortunately boring 3D fractal. All of the interesting fractal nature occurs in the 2D plane, and the rest is just a consequence of rotating that image around the x-axis. We get some interesting fractal artifacts if we slice the rotated Mandelbrot image.

## Motivation

fig 1. Two surfaces normal to the interface.

Most electrodynamics textbooks either start with or contain a treatment of boundary value conditions. These typically involve evaluating Maxwell’s equations over areas or volumes of decreasing height, such as those illustrated in fig. 1, and fig. 2. These represent surfaces and volumes where the height is allowed to decrease to infinitesimal levels, and are traditionally used to find the boundary value constraints of the normal and tangential components of the electric and magnetic fields.

fig 2. A pillbox volume encompassing the interface.

More advanced topics, such as evaluation of the Fresnel reflection and transmission equations, also rely on similar consideration of boundary value constraints. I’ve wondered for a long time how the Fresnel equations could be attacked by looking at the boundary conditions for the combined field $$F = \BE + I c \BB$$, instead of the considering them separately.

## A unified approach.

The Geometric Algebra (and relativistic tensor) formulations of Maxwell’s equations put the electric and magnetic fields on equal footings. It is in fact possible to specify the boundary value constraints on the fields without first separating Maxwell’s equations into their traditional forms. The starting point in Geometric Algebra is Maxwell’s equation, premultiplied by a stationary observer’s timelike basis vector

\label{eqn:maxwellBoundaryConditions:20}
\gamma_0 \grad F = \inv{\epsilon_0 c} \gamma_0 J,

or

\label{eqn:maxwellBoundaryConditions:40}
\lr{ \partial_0 + \spacegrad} F = \frac{\rho}{\epsilon_0} – \frac{\BJ}{\epsilon_0}.

The electrodynamic field $$F = \BE + I c \BB$$ is a multivector in this spatial domain (whereas it is a bivector in the spacetime algebra domain), and has vector and bivector components. The product of the spatial gradient and the field can still be split into dot and curl components $$\spacegrad M = \spacegrad \cdot M + \spacegrad \wedge M$$. If $$M = \sum M_i$$, where $$M_i$$ is an grade $$i$$ blade, then we give this the Hestenes’ [1] definitions

\label{eqn:maxwellBoundaryConditions:60}
\begin{aligned}
\end{aligned}

With that said, Maxwell’s equation can be rearranged into a pair of multivector equations

\label{eqn:maxwellBoundaryConditions:80}
\begin{aligned}
\spacegrad \cdot F &= \gpgrade{-\partial_0 F + \frac{\rho}{\epsilon_0} – \frac{\BJ}{\epsilon_0 c}}{0,1} \\
\end{aligned}

The latter equation can be integrated with Stokes theorem, but we need to apply a duality transformation to the latter in order to apply Stokes to it

\label{eqn:maxwellBoundaryConditions:120}
\begin{aligned}
&=
&=
&=
&=
\end{aligned}

so

\label{eqn:maxwellBoundaryConditions:100}
\begin{aligned}
\spacegrad \wedge (I F) &= I \lr{ -\inv{c} \partial_t \BE + \frac{\rho}{\epsilon_0} – \frac{\BJ}{\epsilon_0 c} } \\
\spacegrad \wedge F &= -I \partial_t \BB.
\end{aligned}

Integrating each of these over the pillbox volume gives

\label{eqn:maxwellBoundaryConditions:140}
\begin{aligned}
\oint_{\partial V} d^2 \Bx \cdot (I F)
&=
\int_{V} d^3 \Bx \cdot \lr{ I \lr{ -\inv{c} \partial_t \BE + \frac{\rho}{\epsilon_0} – \frac{\BJ}{\epsilon_0 c} } } \\
\oint_{\partial V} d^2 \Bx \cdot F
&=
– \partial_t \int_{V} d^3 \Bx \cdot \lr{ I \BB }.
\end{aligned}

In the absence of charges and currents on the surface, and if the height of the volume is reduced to zero, the volume integrals vanish, and only the upper surfaces of the pillbox contribute to the surface integrals.

\label{eqn:maxwellBoundaryConditions:200}
\begin{aligned}
\oint_{\partial V} d^2 \Bx \cdot (I F) &= 0 \\
\oint_{\partial V} d^2 \Bx \cdot F &= 0.
\end{aligned}

With a multivector $$F$$ in the mix, the geometric meaning of these integrals is not terribly clear. They do describe the boundary conditions, but to see exactly what those are, we can now resort to the split of $$F$$ into its electric and magnetic fields. Let’s look at the non-dual integral to start with

\label{eqn:maxwellBoundaryConditions:160}
\begin{aligned}
\oint_{\partial V} d^2 \Bx \cdot F
&=
\oint_{\partial V} d^2 \Bx \cdot \lr{ \BE + I c \BB } \\
&=
\oint_{\partial V} d^2 \Bx \cdot \BE + I c d^2 \Bx \wedge \BB \\
&=
0.
\end{aligned}

No component of $$\BE$$ that is normal to the surface contributes to $$d^2 \Bx \cdot \BE$$, whereas only components of $$\BB$$ that are normal contribute to $$d^2 \Bx \wedge \BB$$. That means that we must have tangential components of $$\BE$$ and the normal components of $$\BB$$ matching on the surfaces

\label{eqn:maxwellBoundaryConditions:180}
\begin{aligned}
\lr{\BE_2 \wedge \ncap} \ncap – \lr{\BE_1 \wedge (-\ncap)} (-\ncap) &= 0 \\
\lr{\BB_2 \cdot \ncap} \ncap – \lr{\BB_1 \cdot (-\ncap)} (-\ncap) &= 0 .
\end{aligned}

Similarly, for the dot product of the dual field, this is

\label{eqn:maxwellBoundaryConditions:220}
\begin{aligned}
\oint_{\partial V} d^2 \Bx \cdot (I F)
&=
\oint_{\partial V} d^2 \Bx \cdot (I \BE – c \BB) \\
&=
\oint_{\partial V} I d^2 \Bx \wedge \BE – c d^2 \Bx \cdot \BB.
\end{aligned}

For this integral, only the normal components of $$\BE$$ contribute, and only the tangential components of $$\BB$$ contribute. This means that

\label{eqn:maxwellBoundaryConditions:240}
\begin{aligned}
\lr{\BE_2 \cdot \ncap} \ncap – \lr{\BE_1 \cdot (-\ncap)} (-\ncap) &= 0 \\
\lr{\BB_2 \wedge \ncap} \ncap – \lr{\BB_1 \wedge (-\ncap)} (-\ncap) &= 0.
\end{aligned}

This is why we end up with a seemingly strange mix of tangential and normal components of the electric and magnetic fields. These constraints can be summarized as

\label{eqn:maxwellBoundaryConditions:260}
\begin{aligned}
( \BE_2 – \BE_1 ) \cdot \ncap &= 0 \\
( \BE_2 – \BE_1 ) \wedge \ncap &= 0 \\
( \BB_2 – \BB_1 ) \cdot \ncap &= 0 \\
( \BB_2 – \BB_1 ) \wedge \ncap &= 0
\end{aligned}

These relationships are usually expressed in terms of all of $$\BE, \BD, \BB$$ and $$\BH$$. Because I’d started with Maxwell’s equations for free space, I don’t have the $$\epsilon$$ and $$\mu$$ factors that produce those more general relationships. Those more general boundary value relationships are usually the starting point for the Fresnel interface analysis. It is also possible to further generalize these relationships to include charges and currents on the surface.

# References

[1] D. Hestenes. New Foundations for Classical Mechanics. Kluwer Academic Publishers, 1999.


text:
# Complex Impedance

1. Mar 30, 2012

### BuddhaBelly34

1. The problem statement, all variables and given/known data
Time-domain expression for i(t), based on measures above:
Time-domain expression for i(t), based on measured R and L:

Vp-p on 16Ω resistor: 7.25mV "measured above"
Ip-p for resistor: 0.676mA "measure above"
Ip for resistor: .338mA "measured above"

R: 14.8Ω L: 10.1mH "measured R and L"

Source voltage is 5Vp-p 0 off-set at 1kHz
2. Relevant equations
? The issue is that I need to make the equations I suppose.

3. The attempt at a solution
This is for the measured R and L values:

Ip = Vp/(R + jωL) = 2.5/(14.8 + j2000∏*10.1) = 2.5/(14.8 + j63.5) = 37/622 - j(159/622) = 59E-3 - j256E-3

i(t) = R(cosωt - β)

R = √(x2 + y2) = √((59E-3)2 + ((256E-3)2) = 263E-3

β = tan-1(-256/59) = -77°

i(t) = 0.263cos(2000∏t + 77°)

Some specific issues:
1. How in the world can i(t) = Rcos(ωt - β)? Wouldn't that make voltage lag current which shouldn't happen when an inductor is in the circuit?

2. When I do the problem with
Vp-p on 16Ω resistor: 7.25mV "measured above"
Ip-p for resistor: 0.676mA "measure above"
Ip for resistor: .338mA "measured above"
should I be using Vp = 7.25E-3/2 or Vp = Vsource p-p/2? The instructions in the lab manual and the slides from the professor that wrote the manual where not clear.

Last edited: Mar 30, 2012
2. Mar 30, 2012

### Staff: Mentor

Is there a particular circuit configuration that we should know about?

3. Mar 30, 2012

### BuddhaBelly34

Lol I'm not sure, but here it is:

PS, this is an intro-lab if that wasn't obvious enough. lol

Last edited: Mar 30, 2012
4. Mar 30, 2012

### Staff: Mentor

You don't want to mix p-p and p quantities. Choose one "type" for your computations and stick to it.

Something misfired in your complex number calcs above; The result looks off.

Also, if I go by the 5Vpp @1KHz source and the given component values, the magnitude of the current that I calculate would be about 77 mA p-p, or 38 mA peak. So something is fishy about your "measured above" values or the component values.
What's a "current voltage"?
Well, it depends upon what it is you're trying to calculate. Make a clear statement of what it is you're trying to determine. If it's the magnitude of the current flowing through the resistor, then Ip = Vp/R, or Ipp = Vpp/R, where both Vp and Vpp are measured across R.

5. Mar 30, 2012

### BuddhaBelly34

Edited my post, I didn't mean "current voltage" I was typing "voltage lag current" but added a current in front of voltage.

As for picking "p-p" or "p" it was just a type-o going to fix that too.

"Also, if I go by the 5Vpp @1KHz source and the given component values, the magnitude of the current that I calculate would be about 77 mA p-p, or 38 mA peak. So something is fishy about your "measured above" values or the component values."
This is something that I, unfortunately, cannot fix. This is my professor's first time teaching this class, so there is more self teaching than anything and I probably made some mistakes with the oscilloscope (which is basically push buttons and hope you get it right :( ).

"Something misfired in your complex number calcs above; The result looks off. " Like rounding errors? Or something in the algebra? I was rounding as I was going.

6. Mar 30, 2012

### BuddhaBelly34

"Well, it depends upon what it is you're trying to calculate. Make a clear statement of what it is you're trying to determine. If it's the magnitude of the current flowing through the resistor, then Ip = Vp/R, or Ipp = Vpp/R, where both Vp and Vpp are measured across R."

I'm assuming it's Ip because the lab manual says "... i(t) = Ipcos(wt-b)".

7. Mar 30, 2012

### Staff: Mentor

$$\frac{2.5\;V}{14.8 + 63.46j \; \Omega} \cdot \frac{14.8 - 63.46j}{14.8 - 63.46j} → \frac{37 - 158.65j}{4246}\;A = 8.71 + 37.36j \; mA$$

Last edited: Mar 30, 2012
8. Mar 30, 2012

### BuddhaBelly34

edit since there is no delete.

Last edited: Mar 30, 2012
9. Mar 30, 2012

### BuddhaBelly34

okay I found my mistake. this guy in my class keeps changing my ti-83's notation to eng from normal and i miss the E's half the time. I get what you get now (now being paying attention to the Es lol)

So if the measured values look wrong should I go in to the lab and remeasure?

Okie doke, phase angle stays the same but now I have R = 0.0387. :P

Last edited: Mar 30, 2012
10. Mar 30, 2012

### Staff: Mentor

That might not be a bad idea. You might want to determine at least the order of magnitudes to expect for the values. Also confirm the part values and supply frequency. If you have an Ohmmeter handy, check to see if the inductor has a non-negligible resistance.

When using an oscilloscope, always check to see if any scope probe you're using has a scaling factor associated with it (quite often they divide by ten).

For the current that you measured, what type of instrument did you use? Was it an ammeter? Does it provide values in peak or RMS? (typically they present RMS values assuming a sinusoidal signal).

Supposing that R = 14.8 Ω, L = 10.1 mH, and f = 1 KHz, then what do you think you should see for the voltage across the resistor?

Last edited: Mar 30, 2012
11. Mar 30, 2012

### BuddhaBelly34

I thought that I was supposed to calculate the current? http://utdallas.edu/~dodge/EE1202/lab5.pdf Top of page 7.

Not sure what I should be seeing across the resistor. I don't understand AC circuits that well. Since it is in series the voltage across the resistor and the inductor must equal the source voltage right? So VL + VR = VS. I think VL = IXL where XL = wL.

Last edited: Mar 30, 2012
12. Mar 30, 2012

### Staff: Mentor

Ah. From the caption on the figure there: "The 'current' trace is really resistor voltage, but it is in phase with the current."

So the actual (p-p) current would be given by that curve's p-p voltage divided by the resistance.

13. Mar 30, 2012

### BuddhaBelly34

Yeah that is why I made the edit. Sorry, too much going on in my head right now. I had that problem first last night then the professor corrected me. Assuming my lab partner took the right reading for voltage on the resistor I have V = 7.25mV.

Ip = Vp-p/2R = 244.9E-6 A

14. Mar 30, 2012

### Staff: Mentor

You can treat the impedance of the inductor just as you would a resistance as far as the usual circuit equations are concerned. The only difference is that you'll be dealing with complex values in the calculations.

Your two components (resistor, inductor) are going to behave like a voltage divider. So if E is the input voltage and R and ZL the impedances of the parts, what voltage should you expect across the R?

15. Mar 30, 2012

### BuddhaBelly34

I'm not sure. Sorry for being ignorant, but there was no "intro to EE" class before this lab course. So I'm learning as I go.

16. Mar 30, 2012

### Staff: Mentor

If you can calculate the (complex) current, then you can determine the (complex) voltages for the individual components via Ohm's law. Their magnitudes should match those of the traces you see on the scope.