## 2.1 Mathematical Relationships for Ratio-Based Expressions in Artificially Created Linear Denominator vs. Numerator Relationships

The first artificial datasets (Fig. 1a-e) demonstrate the effect of very small differences in *y*-intercepts for linear denominator vs. numerator functions on observed 87Sr:86Sr ratios. Plots of the umole/ml denominator vs. the umole/ml numerator of isotopic ratios as presented in Fig. 1a and c are not evaluated in the isotopic literature. They are presented here to demonstrate the mathematical relationships between linear denominator vs. numerator plots and plots of the denominator or total isotope vs. an isotopic ratio. The datasets presented in Fig. 1a-e, represent a system where Sr-containing material with a constant 87Sr:86Sr ratio is being added to or removed from an existing system, where the slope value for an 86Sr vs. 87Sr plot is equal to the 87Sr:86Sr ratio of modern seawater. For simplicity, this artificial data (and similar data in Fig. 1c and d) is first imagined as coming from a system where Sr-containing material with a constant 87Sr:86Sr ratio is being added to an existing system. Slopes of denominator vs. numerator regression equations will therefore be related to sources (the isotopic ratio of an element gained). The same system viewed as losing total element will be subsequently discussed.

With the linear equations as a starting point, it is easier to take the next step and demonstrate that isotopic ratios must depend on their denominators and total isotope concentrations (Fig. 1b and d). The slope of the 86Sr vs. 87Sr regression line is the isotopic ratio of the material being added to the system. This dataset results in a linear relationship when 86Sr vs. 87Sr are plotted because the ratio of 87Sr:86Sr in the source remains constant. In Fig. 1a, the existing system background has a higher 87Sr:86Sr ratio than the material being added. Small deviations from an isometric relationship result in 3 infinitesimally different regression lines, each with very small positive *y*-intercepts. By plotting the more common isotope (86Sr) against 87Sr:86Sr ratio in Fig. 1b, it is possible to visualize the consequences of the “invisible” differences in the 3 regressions from Fig. 1a. In Fig. 1b, the values at the far left represent the background ratios at the starting point. The differences reflected in 87Sr:86Sr ratio (at the 3rd and 4th decimal place) in this artificial example are in the range of many 87Sr:86Sr ratios reported.

As source material is added (which has a lower 87Sr:86Sr ratio than the existing system), values for the 87Sr:86Sr ratio shown in Fig. 2a asymptotically approach the values for the slopes of the regressions in Fig. 1a. This occurs because for all linear functions *y* = m*x* + b, the ratio is defined by *y*/*x* = (1/*x*) b + m. For high values of 86Sr or total Sr the (1/*x*) b term approaches zero, thus the ratio value approaches the original 86Sr vs. 87Sr slope.

In Fig. 1c, the existing system background has a lower 87Sr:86Sr ratio than the material being added, resulting in three distinct functions with very small negative *y*-intercepts. The consequences of these differences are more easily visualized in Fig. 1d, where the values at the far left represent the background ratios at the starting point. Note that these *y*-axis values are therefore generally lower than the values in Fig. 1b. However, in both Fig. 1b and d, the values at the far right are approaching the ratio of 0.7091 found in the imaginary modern seawater source. As source material with a higher 87Sr:86Sr ratio than the background is added, the 87Sr:86Sr ratio increases and asymptotically approaches the slope of the 86Sr vs. 87Sr regression lines in Fig. 1c. Again, this occurs because for all linear functions *y* = m*x* + b; the ratio is defined by *y*/*x* = (1/*x*) b + m. For high values of 86Sr or total Sr the (1/x) b term approaches zero, thus the ratio value, once again, approaches the original 86Sr vs. 87Sr slope.

The relationship between the *y*-intercepts of the 86Sr vs. 87Sr artificial datasets (created to have a slope that is equivalent to modern seawater) from Fig. 1a and c, plotted against the isotopic ratio of the system prior to any addition shows that *y*-intercepts are perfectly related to the backgrounds of the artificial examples (Fig. 1e). The three blue points on the far right in Fig. 1e represent the scenarios for positive *y*-intercepts shown in Fig. 1a. The three blue points on the far left represent the scenarios for negative *y*-intercepts shown in Fig. 1c. The central red point represents an initial 87Sr:86Sr ratio which is the same as the exogenous additions. Under these conditions (source = background) the 87Sr:86Sr ratio will not change, as is demonstrated in the red lines in Fig. 1b and d. A plot of denominator vs. numerator (*y* = m*x* + b) reveals both the source (m) and a value related to background (b) which, like the mathematically related mixing diagram, allows one to interpret isotopic data with respect to sources and backgrounds instead of relying only on observed ratios.

In a review of 24 Sr isotope publications, with over 75 datasets, relationships between ratio denominators and numerators were linear for more than 66% of the cases (data not shown but available on request). In all cases where isotopic ratios changed with time, depth, or treatment, total Sr also changed. Non-zero *y*-intercepts for linear denominator isotope vs. numerator isotope concentration plots (*y* = m*x* + b) are likely ubiquitous.

There are several ways to correct ratio confounding errors related to non-zero *y*-intercepts in denominator vs. numerator plots. One approach is adjusting standard isotopic ratios (always calculated on a molar basis) to remove the background and total element effects that confound interpretation of isotope ratios. The value of a positive *y*-intercept for a denominator vs. numerator plot (in umolar units) can be subtracted from the numerator isotope values of a dataset before “new” ratios are recalculated with the original denominators. Similarly, the absolute value of a negative *y*-intercept can be added to the numerator isotope values of a dataset and then a “new” calculated ratio calculated from the modified numerator and original denominator value.

This will produce “corrected” isotopic ratios where the average of relatively similar ratio values matches the slope of denominator vs. numerator regression lines (Supplemental Material 8; and16). The flat *y*-intercept adjusted mixing lines have the same *y*-intercept as a standard mixing line. Differences in corrected ratios that arise from less than perfect denominator vs. numerator relationships represent source changes for corresponding treatments, sampling time, or depth (Supplemental Material 8; and16).

For linear denominator vs. numerator plots, even with very high r2 values, points do not fall exactly on the regression line. Points slightly above (+ residual) or below (- residual) the line respectively represent a source slightly higher or lower than the regression line slope. These residuals are strongly related to the *y*-intercept modified ratios (data not shown).

The data in Fig. 1a-d) could also be viewed as a system that is losing total element. In this case the 87Sr:86Sr values at the far right of Fig. 1c and d represent the background of systems before any Sr has been removed. Because the observed ratios in Fig. 1b are greater than the loss ratio the system will become more enriched as higher 87Sr:86Sr material is left behind. In Fig. 1d the observed ratios are smaller than the loss ratio, thus the system will become more depleted as lower 87Sr:86Sr material is left behind.

The first scenario (Fig. 1b) mimics isotopic fractionation as total Sr is lost from a system. Chemical or biological discrimination favoring the loss of the lighter isotope could lead to more enrichment of the heavy isotope. However, this increase in 87Sr:86Sr enrichment is caused by a constant loss source that is independent of substrate levels (observed ratios) rather than constant discrimination (defined by a constant fractionation factor) that favours the light isotope. The *y*-intercepts in Fig. 1a and c are also perfectly correlated with the before-loss backgrounds at the far right in Fig. 1b and d. (data not shown). There is a much smaller range of *y*-axis values than what appears in Fig. 1e, but the r2 value is still 1.0.

Denominator vs. numerator slopes for either the artificial (Fig. 1a and c) or real data will define a net source (either loss or gain), but give no hint about whether a system is gaining, losing, or experiencing up and down changes in total element. The changing slopes of curvilinear functions described below have been viewed as representing a system with increasing total element to more simply describe the dynamics of observed ratios. However, these systems could also be viewed as losing total element.

## 2.2 Mathematical Relationships for Ratio-Based Expressions in Artificially Created Curvilinear Denominator vs. Numerator Relationships

Additional artificial datasets were used to investigate curvilinear relationships. The first example was derived from data presented by Grupe *et al.*3 in their evaluation of human bone and teeth. This process, using the best-fit relationship for real data, ensures that the curvatures are realistic. The umole/g units used by Grupe *et al.*3 have been expressed as umole/ml to create a liquid example analogous to Fig. 1.

Concave functions with negative and positive *y*-intercepts were respectively created by subtracting or adding 0.05 moles/ml to the original polynomial predicted 87Sr values. This creates three functions with similar curvature differing only in their *y*-intercepts. The 0.05 moles/ml adjustment is a small portion (~ 0.09%) of the original 87Sr data range (~ 60 umoles/ml), but is sufficient to make an important difference in the 87Sr:86Sr ratios because differences in the third and fourth decimal are commonly evaluated. Equation-derived functions (formulas in Supplemental Material Section 8; Note on adjusting ratios to account for non-zero y-intercepts for polynomial functions) that predict the isotopic ratios for a polynomial equation with a zero *y*-intercept exactly match the zero *y*-intercept corrected isotopic ratios. Additional comments on the importance of very small *y*-intercepts which will apply to either linear or curvilinear denominator vs. numerator functions are presented in Supplemental Material: Section 9; Note on the magnitude of non-zero y-intercepts.

All three functions in Fig. 2a have the same constantly declining derivative (not shown) as total Sr or 86Sr increases, but only for the function with a zero *y*-intercept will derivatives and 87Sr:86Sr ratios provide similar information. The concave polynomial relationship for an 86Sr vs. 87Sr relationship suggests that the source is becoming more depleted with added material. Evaluation of this new set of artificially created examples (Fig. 2a) results in three possible patterns: (A) if the *y*-intercept is zero, the ratio of 87Sr:86Sr will decline as more Sr is added to the system, simply because the source is becoming increasingly depleted. In this case, the original curvature (based on a changing source) is the only driver for changes in the isotopic ratio. (B) If the *y*-intercept is positive (indicating that the original source is less than the background), the ratio of 87Sr:86Sr will decline as more source is added to the system. In this case, the decline in the ratio is most rapid initially where the ratio overrestimates what function curvature suggests. (C) Interestingly, if the *y*-intercept is negative, initially there is an increase in the ratio of 87Sr:86Sr as the total Sr or 87Sr increases. This occurs because the source has a higher ratio than the background. Ultimately as the source becomes more depleted, there is a decline in the 87Sr:86Sr ratio of the system.

Convex functions were created by reversing the sign of the *x*2 term in the original relationship and then negative and positive *y*-intercepts were once again created by subtracting or adding 0.05 umoles/ml to the original 87Sr values. This also creates three functions with similar curvature differing only in their *y*-intercepts. Note that a convex polynomial relationship defined by 86Sr vs. 87Sr plots suggests that the source is becoming more enriched with added material. Evaluation of these artificially created curvilinear datasets (Fig. 2b) results in three different patterns than what are presented in Fig. 2a: (A) if the *y*-intercept is zero, the 87Sr:86Sr ratio will increase as more Sr is added to the system, simply because the source is more enriched than the system itself. In this case, the original 86Sr vs. 87Sr curvature (based on a changing source) is the only driver for changes in the isotopic ratio. (B) If the *y*-intercept is positive (indicating that the original source is less than the background), the ratio of 87Sr:86Sr will initially decrease as more Sr is added to the system, but as the source becomes increasingly enriched, the ratio will begin to increase as the source becomes more influential than the background. (C) If the *y*-intercept is negative, the ratio of 87Sr:86Sr increases as more enriched Sr is added to the system. In this case, the increase in the ratio is most rapid initially where the ratio underestimates what function curvature suggests. Supplemental Material; Section 10 simulations of curvilinear denominator vs. numerator relationships also suggest that the non-zero *y*-intercepts in polynomial functions are perfectly related to system backgrounds.

## 2.3 Quantifying Isotopic Sources

An introduction of isotopic sources is presented in Supplemental Material: Section 5, Quantifying isotopic sources and Supplemental Material and Section 6, More nuances of Keeling plots and mixing diagrams). Here we include source evaluations of material presented in Fig. 1a-e.

A mixing diagram for the data presented in Fig. 1a and c is presented in Fig. 3a. For the 1/denominator (1/86Sr in umole/ml units) form of the mixing diagrams, slopes match the *y*-intercepts shown in Fig. 1a and c. The *y*-intercepts of the mixing diagrams match the slope of the original 86Sr vs. 87Sr plots shown in Fig. 1a and c. As shown above, the *y*-intercepts of 86Sr vs. 87Sr plots are related to backgrounds. Either a mixing diagram (Fig. 3a) or an 86Sr vs. 87Sr plot can identify exogenous sources and relative backgrounds.

A different form of mixing diagram for the data in Fig. 1a and c is presented in Fig. 3b. For this example, the 1/total Sr concentration expression in umole/ml units is used as the *x*-axis rather than the 1/86Sr (in umole/ml units) used in Fig. 3a. For the 1/total element (in umole/ml units) form of the mixing diagram, the slopes do not exactly match the *y*-intercepts shown in Fig. 1a and c. The mixing line slopes match denominator vs. numerator *y*-intercepts if the 1/denominator form of the mixing line is used because the two equations are equivalent *y* = m*x* + b and *y*/*x* = (1/*x*) b + m. The *x* terms differ for these two equations if the 1/total umoles element form (in Fig. 3b) of the mixing line is used. The *y*-intercepts (in Fig. 3a and b) of the mixing diagrams that represents a source for both forms match the slopes of the original 86Sr vs. 87Sr plots shown in Fig. 1a and c. This occurs because the denominator of an isotopic ratio and total element are almost perfectly related because the portion of total for the denominator varies over a miniscule range. The 1/total element values in either ug/ml or umoles/ml units are a surrogate for 1/87Sr in umole/ml units. Conventional mixing diagrams in the isotope literature usually use 1/total Sr concentrations in mass units (ug/ml in Fig. 3c) rather than molar units that are shown in both Fig. 3a and b). As stated above the *y*-intercept of the 1/total element concentration in mass units mixing line form also correctly identifies the source. Only the 1/86Sr form (Fig. 3a) of mixing line (using molar values) produces slopes exactly matching *y*-intercepts in Fig. 1a and c.

Different background isotopic composition can cause mathematical-driven differences in *y*-intercepts for denominator isotope vs. numerator isotope plots. However, just because *y*-intercepts differ, does not necessarily suggest that background differences are the sole cause. The *y*-intercepts of 86Sr vs. 87Sr regression equations are extremely small. Source detection, either from denominator isotope vs. numerator isotope slope; or indirectly from standard Keeling plots or mixing diagram *y*-intercepts, are less prone to error and can be more accurately calculated than background isotopic composition.

Linear denominator isotope vs. numerator isotope functions (*y* = m*x* + b) must produce linear mixing diagrams (*y*/*x* = (1/*x*) b + m). Detectable curvature in a Keeling plot or mixing diagram suggests that the original denominator isotope vs. numerator isotope function is not linear. Therefore, an evaluation for possible Keeling plot or mixing line curvature can supplement denominator vs. numerator evaluations for statistically significant *x*2 terms in polynomial best-fit equations, and spline smoothed cubic derivatives when assessing whether a denominator vs. numerator plot is linear.

## 2.4 Sources, Backgrounds and y-intercepts for Curvilinear Denominator vs. Numerator Functions

The purpose of the simulations (presented in Supplemental Material: Section 10, Simulations demonstrating source, background and *y*-intercept effects on isotopic ratios for curvilinear denominator vs. numerator functions) presented here reiterate the fact that changing observed isotopic ratios are not necessarily related to changing sources. A second simulation goal is to demonstrate that interpreting a nonlinear mixing diagram, although difficult, can still be theoretically used for a changing source system, as data can be manipulated to deal with changing sources. A third goal was to demonstrate that even though *y*-intercepts for curvilinear 86Sr vs. 87Sr relationships in real datasets may be difficult to evaluate, they are mathematically related to system backgrounds.

**2.5 Even if All Treatments in a Study are Defined with Linear Denominator vs. Numerator Functions, Treatments are not Necessarily Similarly Confounded.**

The size of the non-zero *y*-intercept and the distance of data points from the origin affect isotopic ratios’ confounding severity. The effects of different *y*-intercepts and the position of points (both close to and farther away from the origin) can be clearly seen in Fig. 1b and d. The distance from the origin is important because at large values of the denominator a ratio will approach the denominator vs. numerator slope. For treatments with similar slopes and positive denominator vs. numerator *y*-intercepts, an average isotope ratio will be larger for a treatment with many low denominator values than for a treatment with higher denominator values.

The Supplemental Material: Section 11, Simulation demonstrating why not all treatments in an experimental study are similarly confounded outlines how approximated quantitative numerical indicators of the degree of isotopic confounding can be theoretically defined. This is done by combining both *y*-intercept and denominator size effects. A similar real data example from Moyo *et al.*25 reveals that 15N:14N ratios are overestimated for all species in a trophic level study16 Supplemental Material 2. While all six species have positive *y*-intercepts for linear 14N vs. 15N plots, scaling-related overestimates are severe for only two of six species evaluated.