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Isotope notation
In order to study trace metal stable isotope biogeochemistry, it is necessary to compare the relative abundances of isotopes of trace metals in a given biological, geological, or chemical pool and monitor how those relative abundances change as a result of various biogeochemical processes. Conventional notations used to mathematically describe isotope abundances, as exemplified here for 56Fe, include the isotope ratio (56R), fractional abundance (56F) and delta notation (δ56Fe). Furthermore, as different biogeochemical processes vary the relative abundances of the isotopes of a given trace metal, different reaction pools or substances will become enriched or depleted in specific isotopes. This partial separation of isotopes between different pools is termed isotope fractionation, and can be mathematically described by fractionation factors α or ε (which express the difference in isotope ratio between two pools), or by "cap delta" (Δ; the difference between two δ values). For a more complete description of these notations, see the isotope notation section in Hydrogen isotope biogeochemistry.

Isotope ratio and fractional abundance
The relative abundances of isotopes are calculated as either the isotope ratio (xR) or the fractional abundance (xF), which are defined as:

$$^iR= \frac{^iE}{^jE}$$

and

$$^iF = \frac{^iE}{^iE + {}^jE}$$

where iE and jE are abundances of isotopes i and j of an element. By convention, the denominator of the isotope ratio (jE) always contains the most abundant isotope, which is the lighter isotope (e.g., 13C/12C). Fractional abundances are essentially the mole fraction of an isotope in a given pool. Multiplication of the fractional abundance by 100 yields atom percent.

Delta (δ) notation
Isotope ratios of a sample are conventionally reported in delta notation as the relative difference in isotope ratio between a sample and standard of a known isotopic composition:

$$\delta^{i}E = \frac{^{i}R_{sample}}{^{i}R_{std}} -1$$

Delta values are customarily reported in units of per mil (‰), which is obtained by multiplying the above equation by 1000.

Measures of fractionation
The study of trace metal stable isotope biogeochemistry is based on the fact that different biogeochemical processes will vary the relative abundances of the isotopes of a given trace metal, leading to enrichment or depletion of specific isotopes between the different reaction pools or substances. This partial separation of isotopes is termed isotope fractionation. Various method are employed to describe the isotopic fractionation between different pools. One approach uses the fractionation factor (α), which is defined as the difference in isotope ratio between pools A and B:

$$\alpha_{A/B}=\frac{{}^i\!R_A}{{}^i\!R_B}$$

In this equation iRA and iRB are each calculated with respect to a standard of known isotopic composition. Another measure of fractionation is epsilon (ε), which is defined as:

$$\epsilon_{A/B}=\alpha_{A/B}-1$$

As with delta values, epsilon values are commonly reported in units of ‰. The final common measure of fractionation is Δ ("cap delta"), which is defined as:

$$\Delta_{A/B}=\delta^iE_A-\delta^iE_B$$

Stable isotopes and natural abundances
Copper has two naturally occurring stable isotopes: 63Cu and 65Cu, which exist in natural abundances of 69.17 and 30.83%, respectively. The isotopic composition of Cu is reported in delta notation (in ‰) relative to a NIST SRM 976 standard:

$$\delta^{65}Cu = \left [ \frac{(^{65}Cu/^{63}Cu)_{sample}}{(^{65}Cu/^{63}Cu)_{NIST976}} \right ]$$

Chemistry
Copper has two redox states: Cu1+ and Cu2+. The coordination chemistries conferred by its electronic configurations enable Cu to participate in many biological and chemical reactions. In its atomic state, Cu has an electronic configuration of [Ar]3d10 4s1. Upon oxidation to Cu1+, one electron is removed from the 4s shell, giving an electronic configuration of [Ar]3d104s0. Upon further oxidation to Cu2+, an electron is removed from the 3d shell, giving an electronic configuration of [Ar]3d94s0. Due to its full d-orbital, Cu1+ has diamagnetic resonance. In contrast, Cu2+ has one unpaired electron in its d-orbital, giving it paramagnetic resonance.

Equilibrium isotope fractionation
Transitions between redox species Cu1+ and Cu2+ fractionate Cu isotopes. 63Cu2+ is preferentially reduced over 65Cu2+, leaving the residual Cu2+ enriched in 65Cu. The equilibrium fractionation factor for speciation between Cu2+ and Cu1+ (αCu(II)-Cu(I)) is 1.00403 (i.e., dissolved Cu2+ is enriched by ~4‰ relative to Cu1+).

Biology
Copper can be found in the active sites of most enzymes that catalyze redox reactions (i.e., oxidoreductases), as it facilitates single electron transfers while reversibly oscillating between the Cu1+ and Cu2+ redox states. Enzymes typically contain between one (mononuclear) to four (tetranuclear) copper centers, which enable enzymes to catalyze different reactions. These copper centers are generally coordinated to N-, O- and S-containing groups, including histidine, aspartic acid, glutamic acid, cysteine and methionine. Copper's powerful redox capability makes it critically important for biology, but comes at a cost: Cu1+ is a highly toxic metal to cells because it readily abstracts single electrons from organic compounds and cellular material, leading to production of free radicals. Thus, cells have evolved specific strategies for carefully controlling the activity of Cu1+ while exploiting its redox behavior.

Examples of copper-based enzymes
Copper proteins function as electron or oxygen carriers, oxidases, mono- and dioxygenases, superoxide dismutases (SOD) and nitrogen oxide (NOx) reductases. In mollusks, copper-containing hemocyanins bind and transport O2 through the blood, much like hemoglobin in humans. One of the three major families of SOD, which transforms the radical O2- into O2 and H2O2, contains Cu and Zn as its metal cofactors. Additionally, metallothionein, an enzyme that transports toxic metals throughout the cytoplasm, is a Cu-Zn protein. Cytochrome oxidase, which is located in the inner mitochondrial membrane and plays a key role in aerobic respiration by facilitating electron transport to O2, is an Fe-Cu enzyme. In E. coli, the copper protein multicopper oxidase CueO oxidizes toxic Cu1+ to Cu2+ to regulate copper homeostasis. Particulate methane monooxygenase (pMMO) is a copper-containing protein in methanotrophs that oxidizes methane to methanol for both energy and carbon acquisition. Methane oxidation by pMMO is kinetically faster than by soluble methane monooxygenase (sMMO), the iron-containing counterpart to pMMO.

Biological fractionation
Biological processes that fractionate Cu isotopes are not well-understood. The natural 65Cu/63Cu varies according to copper's redox form and the ligand to which copper binds. Oxidized Cu2+ preferentially coordinates with hard donor ligands (e.g., N- or O-containing ligands), while reduced Cu(I) preferentially coordinates with soft donor ligands (e.g., S-containing ligands). As 65Cu is preferentially oxidized over 63Cu, these isotopes tend to coordinate with hard and soft donor ligands, respectively. Cu isotopes can fractionate upon Cu-bacteria interactions from processes that include Cu adsorption to cells, intracellular uptake, metabolic regulation and redox speciation. Fractionation of Cu isotopes upon adsorption to cellular walls appears to depend on the surface functional groups that Cu complexes with, and can span positive and negative values. Furthermore, bacteria preferentially incorporate the lighter Cu isotope intracellularly and into proteins. For example, E. coli, B. subtilis and a natural consortia of microbes sequestered Cu with apparent fractionations (ε65Cu) ranging from ~-1.0 to -4.4‰. Additionally, fractionation of Cu upon incorporation into the apoprotein of azurin was ~-1‰ in P. aeruginosa, and -1.5‰ in E. coli, while ε65Cu values of Cu incorporation into Cu-metallothionein and Cu-Zn-SOD in yeast were -1.7 and -1.2‰, respectively.

Geochemistry
The concentration of Cu in Bulk Silicate Earth is ~30 ppm, slightly less than its average concentration (~72 ppm) in fresh mid-oceanic ridge basalt glass. Cu1+ and Cu2+ form a variety of sulfides (often in association with Fe), as well as carbonates and hydroxides (e.g., chalcopyrite, chalcocite, cuprite and malachite). In mafic and ultramafic rocks, Cu tends to be concentrated in sulfidic materials. In freshwater, the predominant form of Cu is free Cu2+; in seawater, Cu complexes with carbonate ligands to form CuCO3 and Cu(CO3)2]2-.

Cu concentrations in the marine environment
Additionally, Cu is strongly cycled in the surface and deep ocean. Cu concentrations are ~5 nM in the deep Pacific and ~1.5 nM in the deep Atlantic. The deep/surface ratio of Cu in the ocean is typically <10. Depth concentration profiles for Cu are roughly linear due to biological recycling and scavenging processes in addition to adsorption to particles. Similarly, δ65Cu values in the Atlantic ocean do not markedly vary with depth.

Measurement
Cu is first liberated from its host rock (via dissolution with hydrofluoric acid) or from biological material (via digestion with HNO3). Seawater samples must be concentrated due to the low (nM) concentrations of Cu. The sample material is subsequently run through an anion-exchange column, and Cu is isolated. This step can also introduce Cu isotope fractionation if Cu is not quantitatively recovered from the column. Extraction must be quantitative in order to prevent isotope fractionation from occurring at this step. If samples are from seawater, other ions (e.g., Na+, Mg2+, Ca2+) must be removed in order to eliminate isobaric interferences. Prior to 1992, 65Cu/63Cu ratios were measured via thermal ionization mass spectrometry (TIMS). Today, Cu isotopic compositions are measured via multi-conductor inductively coupled plasma mass spectrometry (MC-ICP-MS), which ionizes samples using inductively coupled plasma and introduces smaller errors than TIMS.

δ65Cu values in the terrestrial environment
To first order, δ65Cu values in organisms are driven by dietary Cu isotopic compositions. In plants, δ65Cu values vary between the different components (seeds, stem and leaves) from -1 to +0.4‰. In animals, δ65Cu values vary among the different organs. δ65Cu values of livers from sheep and mice fed a diet of δ65Cu = 0‰ were -1.5‰, while δ65Cu values of their kidneys were +1.5‰ [Balter and Zazzo, 2011]. Serum in human blood is typically 65Cu-depleted relative to erythrocytes, with these blood components having a range of Cu isotopic compositions from -0.7 to +0.9‰. Human muscles have δ65Cu values near 0‰.

δ65Cu values in rocks and minerals
In general, igneous processes do not appear to strongly fractionate Cu isotopes, and the δ65Cu values of mid-oceanic ridge basalts fall around 0‰. The Cu isotope compositions of copper-containing minerals vary over a wide range, likely due to alteration of the primary high-temperature deposits ; other refs. Chalcopyrite from mafic igneous rocks had δ65Cu values of -0.1 to -0.2‰ [ref?], while Cu minerals in black smokers (chalcopyrite, bornite, covellite and atacamite) exhibited a wider range of δ65Cu values from -1.0 to +4.0‰ [ref?]. Additionally, atacamite lining the outer rims of black smokers can be up to 2.5‰ heavier than chalcopyrite contained within the black smoker. δ65Cu values of Cu minerals (including chrysocolle, azurite, malachite, cuprite and native copper) in low-temperature deposits varied widely over a range of -3.0 to +5.6‰.

δ65Cu values in the ocean
Due to equilibrium and biological processes that fractionate Cu isotopes in the marine environment, the bulk isotopic composition of copper (δ65Cu = +0.6 to +1.5‰) is different from the δ65Cu values of the riverine input (δ65Cu = +0.02 to +1.45‰, with discharge-weighted average δ65Cu = +0.68‰) to the oceans. Equilibrium processes that fractionate Cu isotopes include high temperature ion exchange and redox speciation between mineral phases, and low temperature ion exchange between aqueous species or redox speciation between inorganic species. In riverine and marine environments, 65Cu/63Cu ratios are driven by preferential adsorption of 63Cu to particulate matter and preferential binding of 65Cu to organic complexes. The δ65Cu values of a 760 cm sedimentary core taken from the Central Pacific ocean were lighter than the bulk ocean and varied from -0.94 to -2.83‰. δ65Cu values of the surface layers of FeMn-nodules are fairly homogenous throughout the oceans (average = 0.31‰), suggesting low biological demand for Cu in the marine environment compared to that of Fe or Zn. However, Cu isotope compositions of material collected on sediment traps at depths of 1000 and 2500 m in the central Atlantic ocean show seasonal variation with heaviest δ65Cu values in the spring and summer seasons. This suggests seasonal preferential uptake of 63Cu by biological processes.

Medicine
Copper isotopes in human plasma may serve as a marker for various types of cancer. In general, the serum of patients with colon, breast and liver cancer appear to be 65Cu-depleted relative to the serum of healthy patients, while liver tumors are 65Cu-enriched. In one study, the blood of patients with hepatocellular carcinomas was found to be depleted in 65Cu by 0.4‰ relative to the blood of non-cancer patients.

Stable isotopes and natural abundances
Zinc has five stable isotopes: 64Zn, 66Zn, 67Zn, 68Zn and 70Zn, with natural abundances of 48.63, 27.90, 4.10, 18.75, and 0.62%, respectively. The isotopic composition of Zn is reported in delta notation (in ‰):

$$\delta^{x}Zn = \left [ \frac{(^{x}Zn/^{64}Zn)_{sample}}{(^{x}Zn/^{64}Zn)_{std}} \right ]$$

where xZn is commonly either 66Zn or 68Zn. Standard reference materials used for Zn isotope measurements are commonly JMC 3-0749C, NIST-SRM 683 or NIST-SRM 682.

Chemistry
Zinc has only one redox state: Zn2+. In its atomic state, Zn has an electronic configuration of [Ar]3d104s2. Zn2+ has an electronic configuration of [Ar]3d104s0.

Biology
Though zinc is not redox-active, as it only has one redox state, it plays a key role in numerous enzymatic processes. In the cytoplasms of cells, Zn concentrations typically range from a few ppt to a 10s of ppm.

Examples of zinc-based enzymes
Zn is present in the active sites of most hydrolytic enzymes, and is used as an electrophilic catalyst to activate water molecules that ultimately hydrolyze chemical bonds. Both superoxide dismutase (SOD) and metallothionein are Cu-Zn enzymes that reside in the cytoplasm of cells. Carbonic anhydrase, which catalyzes the interconversion of CO2(g) and HCO3-(aq), is a Zn protein. In eukaryotic cells, Zn finger proteins interact with DNA, RNA and proteins to regulate a variety of cellular processes, including DNA repair, transcription and protein degradation. Additional examples of Zn proteins include alcohol dehydrogenase (which oxidizes alcohols) and carboxypeptidase (which hydrolyzes C-terminal peptide residues).

Biological fractionation
Relatively little is known about isotopic fractionation of zinc by biological processes. Many organisms, including certain species of fish, plants and marine phytoplankton, have both high- and low-affinity Zn transport systems, which appear to fractionate Zn isotopes differently. A study by John et al. observed apparent isotope effects associated with Zn uptake by the marine diatom Thalassiosira oceanica of -0.2‰ for high-affinity uptake (at low Zn concentrations) and -0.8‰ for low-affinity uptake (at high Zn concentrations). Additionally, in this study, unwashed cells were enriched in 65Zn, indicating preferential adsorption of 65Zn to the extracellular surfaces of T. oceanica. Results from John et al. demonstrating apparent discrimination against the heavy isotope (66Zn) during uptake conflict with results Gélabert et al. in which marine phytoplankton and freshwater periphytic preferentially uptook 66Zn from solution. The authors explained these results as due to a preferential partitioning of 66Zn into a tetrahedrally coordinated structure (i.e., with carboxylate, amine or silanol groups on or inside the cell) over an octahedral coordination with six water molecules in the aqueous phase, consistent with quantum mechanical predictions. These results may have important implications for Zn isotope variations in the ocean. Isotopic discrimination of Zn varies in different components of higher plants, likely due to the various processes involved in Zn uptake, binding, translocation, etc. For example, Weiss et al. observed heavier δ66Zn values in the roots of several plants (rice, lettuce and tomato) relative to the bulk solution in which the plants were grown. Furthermore, the shoots of the plants were 66Zn-depleted relative to both the roots and bulk solution.

Geochemistry
The concentration of Zn in Bulk Silicate Earth is ~55 ppm, while its average concentration in fresh mid-oceanic ridge basalt glass is ~87 ppm. Like Cu, Zn commonly associates with Fe to form a variety of zinc sulfide minerals such as sphalerite. Additionally, Zn associates with carbonates (e.g., to form smithsonite) and hydroxides. In mafic and ultramafic rocks, Zn tends to concentrate in oxides such as spinel and magnetite. In freshwater, Zn predominantly complexes with water to form an octahedrally coordinated aqua ion [Zn(H2O)6]2+. In seawater, Cl- ions replace up to four water molecules in the Zn aqua ion, forming [ZnCl(H2O)5]+, [ZnCl2(H2O)4]0 and [ZnCl4(H2O)2]-.

Zn concentrations in the marine environment
Zn is an essential biological nutrient in the oceans, and its concentration is largely controlled by uptake by phytoplankton and remineralization. Indeed, Zn has been cited as a limiting nutrient for phytoplankton in photic zones, and thus its concentration in surface waters serves as one control on primary productivity. Zn concentrations are extremely low in the surface ocean (<0.1 nM) but are maximal at depth (~2 nM in the deep Atlantic; ~10 nM in the deep Pacific), indicating a deep regeneration cycle. The deep/surface ratio of Zn is typically on the order of 100, significantly larger than observed for Cu. Additionally, Zn associates with carbonate shells of foraminifera and siliceous frustules in diatoms.

Measurement
The analytical pipeline for preparation of sample material for Zn isotope measurements is identical to that of Cu. As with Cu, Zn isotope compositions of materials were formerly measured via TIMS, but today they are measured via MC-ICP-MS. Importantly, because 64Ni has the same mass as 64Zn, any interference of 64Ni on measurements of Zn isotopic compositions must be corrected for, which is often achieved by measuring one of the isotopes of Ni.

δ66Zn values in rocks and minerals
Fractionation of Zn isotopes by igneous processes is generally insignificant, and δ66Zn values of basalt fall within the range of +0.2 to +0.3‰. δ66Zn values of clay minerals from diverse environments and of diverse ages have been found to fall within the same range as basalts, suggesting negligible fractionation between the basaltic precursors and sedimentary materials. Carbonates appear to be more 66Zn-enriched than other sedimentary and igneous rocks. For example, the δ66Zn value of a limestone core taken from the Central Pacific was +0.6‰ at the surface and increased to +1.2‰ with depth. The Zn isotopic compositions of various ores are not well-characterized, but smithsonites and sphalerites (Zn carbonates and Zn sulfides, respectively) collected from various localities in Europe had δ66Zn values ranging from -0.06 to +0.69‰, with smithsonite potentially slightly heavier by 0.3‰ than sphalerite.

δ66Zn values in the ocean
As seen with copper isotopes, the bulk isotopic composition of zinc in the oceans (δ66Zn = +0.5‰) is heavier than that of the riverine input (δ66Zn = +0.3‰), due to equilibrium and biological processes that fractionate Zn isotopes. δ66Zn values are variable in the upper ~100 m of the water column and increase with depth. A vertical profile of δ66Zn values in a 10 m long sedimentary core taken from the Central Pacific showed no isotopic variation (δ66Zn = 0.2‰), suggesting diagenesis does not significantly fractionate Zn isotopes. Settling material collected from sediment traps at 1000 and 2500 m depths in the central Atlantic ocean over the course of a year had slightly heavier δ66Zn values (by ~+0.20‰) in the spring and summer seasons, suggesting preferential removal of 64Zn by primary productivity in the photic zone. Consistent with this, δ66Zn values of phytoplankton in the ocean are ~+0.16‰, isotopically lighter than that of the bulk ocean. The surface layers of FeMn-nodules are 66Zn enriched at high-latitudes (average δ66Zn = +1‰), while δ66Zn values of low-latitude samples are smaller and more variable (spanning +0.5 to +1‰). This observation has been interpreted as due to high levels of Zn consumption and preferential uptake of 64Zn above the seasonal thermocline at high latitudes during warmer seasons, and transfer of this heavy δ66Zn signal to the settling sedimentary Fe-Mn hydroxides.

Paleoproxy for ancient oceans
Zn has been suggested as a proxy for CO3 of the ancient oceans... [Marchitto et al., 2000]. By accurately measuring the magnitude of biological Zn isotope fractionation, changes in the isotopic composition of Zn in seawater or ancient sediments can be directly related to the extent of biological Zn uptake in surface waters.

Medicine
Zn isotopes appear promising as a tracer for breast cancer. Relative to non-cancerous patients, breast cancer patients are known to have significantly higher concentrations of Zn in their breast tissue, but lower concentrations in their blood serum and erythrocytes, due to overexpression of Zn transporters in breast cancer cells. Consistent with these body-wide shifts in Zn homeostasis, δ66Zn values in breast cancer tissues are anomalously light (varying from -0.6 to -0.9‰) relative to breast tissue of healthy patients (δ66Zn = -0.3 to -0.5‰). However, δ66Zn values of blood and serum were not found to be significantly different between cancerous and non-cancerous patients, suggesting a currently unknown isotopically heavy pool of Zn must exist in cancer patients.