Quality Data from Messy Spectra: How Isometric Points Increase Information Content in Highly Overlapping Spectra

Spectroscopic techniques are immensely useful for obtaining information about chemical transformations while they are happening. However, such data are often messy, and it is challenging to extract reliable information from them without careful calibrations or internal standards. This short introductory review discusses how isometric points (points in a spectrum where the signal intensity remains constant throughout the progress of a chemical transformation) can be used to derive high‐quality data from messy spectra. Such analyses are helpful in a variety of (bio‐)chemical settings, as selected case studies demonstrate.


What Are Isometric Points?
Although spectroscopic analyses often focus on areas of a spectrum where great signal changes happen, areas -or points -where nothing happens can be equally instructive and insightful. [6,7] By definition, such isometric points are specific points in a spectrum where the signal remains constant throughout the progress of a chemical transformation (Figure 1a). Isometric points (from the Greek iso, meaning "equal", and metrikos, meaning "relating to the measure") arise when a physicochemical property one is examining remains unchanged when going from the starting material(s) to the product(s) of a reaction. While the underlying principles of isometric points are shared across different spectroscopic techniques, the nature and nomenclature of the resulting isometric points can vary significantly.
Isosbestic (from the Greek sbestos, "extinguishable") points occur in UV/vis spectroscopy when the starting material and product of a reaction have the same extinction coefficient at a given wavelength. Of all isometric points, isosbestic ones are the most common and most widely known. Isosbestic points are frequent because they only require the absorption spectra of the starting material and the product to intercept. For instance, the UV absorption spectra of the nucleobase 1 b and its corresponding monoanion 1 b À (acquired at identical concentrations) intercept at 286 nm. [9] Thus, absorption spectra of 1 b at different pH values (essentially following the deprotonation of 1 b) will exhibit a constant extinction at 286 nm while the remainder of the spectrum will reflect that of 1 b at lower pH values and that of 1 b À at higher pH values (Figure 1b).
Similarly, IR spectra obtained during a transformation can feature isosbestic points when the absorbance (or transmittance) at one wavelength remains constant. Isosbestic points in IR spectra are less frequently found in the literature because IR spectra tend to be more dynamic and feature higher order signal overlaps, particularly in the fingerprint area. Nonetheless, examples can be found throughout the chemical sciences. For instance, the organocatalytic α-selenation of isovaleraldehyde with the selenophthalimide 2 yields IR spectra overlapping at 858 and 866 cm À 1 in the second derivative, with the disappearance of the aromatic CÀ H vibration peaks of 2 reflecting its conversion to the selenoether 2* (Figure 1c). [11] In fluorescence spectroscopy, isoemissive (from the Latin emitto, "sending out") points can occur when both partners of a reaction emit the same total intensity of light at a given wavelength upon excitation at a given (typically lower) wavelength. IUPAC also defines isolampsic (from the Greek lampein, "to emit light") and isostilbic (from the Greek stilbein, "to shine") as synonyms. [17] True isoemissive points are much rarer than isosbestic points because fluorescence responses are typically not perfectly linear as samples can absorb some of the emitted light when fluorescence and absorption spectra of a given compound (or compound pair) overlap. However, pseudoisoemissive points are not uncommon and do provide useful insights. For instance, the enzymatic reduction of the fluorogenic probe 3 exhibits overlapping (or very close to overlapping) fluorescence spectra, which reflect the formation of the corresponding secondary amine 3* (Figure 1d). [13] Other isometric points beyond these are conceivable and do occur. Although far less common than in other spectroscopic techniques, even NMR spectra can feature close to constant intensity at one chemical shift value. Following the analogy, such points could be referred to as (pseudo-)isointensic points. For instance, the enzymatic hydrolysis of adenosine triphosphate (4), yielding the corresponding diphosphate 4*, gives 1 H NMR spectra exhibiting a pseudo-isointensic point at around 8.37 ppm (Figure 1e). [15] How Does an Isometric Point Help Me?
Isometric points increase the amount of information available from convoluted experimental spectra. Rather than having to rely on absolute changes in signal intensity over time, isometric points enable one to extract quantitative information about the extent of conversion of a transformation directly from messy experimental spectra. [18] This approach builds on the linear combination of spectra of individual species present during a transformation. For instance, UV absorption spectra of mixtures of two compounds result in mixed spectra which are equivalent to the linear combination of their respective individual absorption spectra (Figure 1a). Thus, the shape of mixed spectra can directly be traced back to the composition of said mixture when one has reference spectra of the pure compounds. While computational unmixing of such spectra has become an immensely powerful approach, [19][20][21][22][23] simple normalization to the isometric point is in many scenarios sufficient to generate spectra where analysis of a single reference point generates useful information.
The enzymatic phosphorolysis of the nucleoside 5-bromouridine (1 a) is perhaps a particularly illustrative example. [9] The transformation between 1 a and its corresponding nucleobase 1 b goes in hand with a change of the electronic properties of the chromophore, which grants 1 a and 1 b slightly different UV absorption spectra. Under moderately alkaline conditions, both compounds primarily exist as the pyrimidinolates 1 a À and 1 b À (Figure 2a1) and mixtures of the two will exhibit spectral shapes that arise from an overlap and linear combination of their individual, quite distinct spectra (Figure 2a2 and 2a3). However, at pH 9, 1 a À and 1 b À possess the same extinction coefficient at 288 nm ( Figure 2a2). Thus, any equally concentrated mixture of the two will exhibit the same absorbance at 288 nm. Similarly, during a reaction between the two, the absorbance at 288 nm will remain constant, regardless of the extent of conversion ( Figure 2a4). Therefore, one can choose an arbitrary wavelength other than 288 nm to derive a degree of conversion from any experimental spectrum of this transformation. In most practical scenarios, it makes sense to pick a wavelength where the greatest spectral change (Δɛ) happens, which in this example is 304 nm (one may note that 304 nm is not an absorption maximum of either compound, but rather the wavelength with the greatest difference of their extinction coefficients, Figure 2a2). Given that 1 a À possesses a normalized absorbance (A 304/288 = A 304 /A 288 ) of 0.15 and 1 b À a normalized absorbance of 0.97, all experimental values for the 1 a À !1 b À transformation must lie between those two extremes. Indeed, any change between those two reference values is directly proportional to the change of the molar fraction of either compound. For instance, an A 304/288 of 0.16 will correspond to around 99 % 1 a À (and 1 % 1 b À ) and an A 304/288 of 0.39 will correspond to around 70 % 1 a À (and 30 % 1 b À ). Thus, any conversion of 1 a À to 1 b À will result in a predictable change of the observed A 304/288 ( Figure 2a3). This provides direct access to the degree of conversion as expressed by Equation (1) (Figure 2a5).
To generalize from this example, we may refer to the A 304/288 of the 1 a À !1 b À transformation as an isometrically normalized signal intensity I ϕ/ψ . Thus, a system with an isometric point at ψ can be described with [Eq. (1)]: where d is the molar distribution of the starting material E and the product P (which is equivalent to conversion, with d = 0 meaning 100 % E and d = 1 meaning 100 % P), I ϕ/ψ is the experimentally obtained isometrically normalized signal intensity (signal intensity at the arbitrary ϕ divided by the signal intensity at the isometric point ψ) and I ϕ/ψ,E and I ϕ/ψ,P are the isometrically normalized signal intensities of pure E and pure P. All of these values need to be corrected for background signals and all are by definition dimensionless. Applying Equation (1) to experimental spectra obtained during the 1 a À !1 b À transformation yields reaction course data (here, I ϕ/ψ is derived from absorbances; Figure 2a5). Applying the same equation to experimental absorbance spectra obtained for 1 b in equilibrium with its anion 1 b À at different pH values [24] yields a Boltzmann-type distribution of values which can be fitted to obtain the pK a value of 1 b (Figure 1b). Applying the same equation to the experimental fluorescence spectra obtained during the reduction of the fluorogenic probe 3 also yields reaction courses (in this example I ϕ/ψ is derived from Figure 1. Isometric points observed during various transformations and information content available from the associated spectroscopic data. The mathematical operations for each dataset are elaborated upon in the reference section. The data in a belong to 5-nitrouracil and its monoanion (previously unpublished data) and serve as an example of an arbitrary transformation. [8] The data in b were taken from our recent publication about phosphate detection with the PUB module. [9,10] The data in the remaining panels were originally published by Hutchinson, Welsh and Burés (c), [11,12] Lei and colleagues (d), [13,14] and Bramham, Zalar, and Golovanov (e), [15,16] and I am grateful to the authors for generously providing the raw data upon request. Please see their original publications for the structure of the organocatalyst and the R, R', and Ar residues. The nucleobase NB is adenine.

Figure 2.
Case studies for the use of isometrically normalized signal intensities. a) Normalization to the isosbestic point (288 nm) enables quantitative reaction monitoring through the analysis of absorption changes (for instance) at 304 nm. Without having to rely on extinction coefficients, parallel monitoring at 288 and 304 nm returns quantitative conversion data via Equation (1). These data were taken from our recent publication about phosphate detection with the PUB module.
[9] b) Side reactions that create additional observable compounds lead to reaction systems to which isometric normalization cannot be applied. During the phosphorolysis of 5 a at pH 9, four UV-active species are present in equilibrium, which yields non-intercepting absorption spectra. This behavior cannot be accounted for through signal corrections. These data were taken from our short report on alternative assay reagents for PUB. [28] c) The relatively slow glycosylation of 5-nitrouracil shows no real isosbestic point as the path length is not constant; this can easily be accounted for through normalization to either isosbestic point. This reaction may serve as an example for how isometric normalization can correct for routine changes in signal intensity. These are previously unpublished data.

ChemBioChem
Concept doi.org/10.1002/cbic.202200744 fluorescence intensities; Figure 1d). Notably, none of these examples require any previous calibration of equipment or knowledge of extinction coefficients or quantum yields. [25] An analysis of isometrically normalized signal intensities only demands the presence of a (pseudo-)isometric point as well as knowledge of the reference values I ϕ/ψ,E and I ϕ/ψ,P for the starting material and product of the transformation (both of which are available from a single measurement each or can be fitted from known distributions). [16] In analogy, these principles can be extended to any situation where isometric points arise. In addition to the reaction courses and protonation equilibria shown herein, analogous analyses are conceivable for titration experiments with isometric points [26] or electrochemical studies. [27] As such, isometrically normalized signal intensities provide an immensely straightforward way to access highquality data from convoluted experimental spectra.

What if There Is No Isometric Point?
Although many transformations exhibit isometric points, not every reaction will offer itself up for analysis with isometrically normalized signal intensities. Some transformations inherently do not have a useful isometric point. If changes to a chromophore lead to either reactant being UV-inactive or nonfluorescent or if an isometric point has a signal intensity which is near the baseline (or otherwise highly convoluted by noise), one cannot calculate meaningful values for I ϕ/ψ . Reactions involving more than two observable reactants might not have an isometric point either, although such scenarios are conceivable for three reactants. [29,30] For many reactions, the choice of reaction conditions will dictate if there is an isometric point and if it is useful for analysis, as physicochemical properties can vary greatly depending on pH, temperature and solvent. However, given the absence of well curated literature data and lack of thoroughly described case studies on the matter, such effects currently remain to be approached by trial and error and on a case-to-case basis. Almost counterintuitively, the lack of an isometric point can sometimes be just as insightful as the presence of one. Generally, if the absorption or fluorescence spectra of the starting material and product of a given reaction (at the same or at least similar concentrations) intersect, there should be an isometric point. Thus, if one would expect an isometric point to occur based on the available reference spectra, the absence of an isometric point in an experiment usually indicates an imperfect transformation. [31] Side reactions which drain material from the mass balance of a transformation can easily lead to signals at isometric points not giving constant signal intensities. This can include instability of either reactant, further transformations of products, precipitation of reactants or secondary equilibria engaging either reactant. For instance, contrary to the phosphorolysis of bromouridine (1 a), the phosphorolysis of iodouridine (5 a) does not possess an isosbestic point at pH 9. [28] Iodouridine and its nucleobase 5 b have slightly higher pK a values than 1 a and 1 b. Therefore, at pH 9, there exist relevant equilibria between the neutral species 5 a and 5 b and their anions 5 a À and 5 b À (Figure 2b1). However, since the pK a of 5 a differs by that of 5 b by 0.18 pH units (Figure 2b4), the (de-)protonation state does not remain constant during the (5 a + 5 a À )!(5 b + 5 b À ) transformation (Figure 2b5). Thus, at any given point during this transformation after reaction initiation, four UV active species will be present in non-constant concentrations and non-constant ratios. This results in UV absorption spectra without an isosbestic point (Figure 2b2 and  2b3). This issue can be resolved by recording spectra at higher pH values where 5 a and 5 b exclusively exist as their anions and only two observable reactants (5 a À and 5 b À ) are present. [19] Somewhat similarly, the glycosylation of 5-nitrouracil (6 b) only shows a pseudo-isosbestic point under conditions where it should -intuitively -exhibit a perfect isosbestic point (Figure 2c1 and 2c2). In this case, the experimental spectra were obtained over 4 h in an uncovered multiwell plate as this transformation proceeds relatively slowly. The observation of lower signal intensities at what should be an isosbestic point (318 nm) is caused by a shortening of the beam path through the reaction mixture, as a result of the reaction mixture running up the sides of the well. Thus, the total observed signal intensity drops throughout the progress of the reaction. However, normalization to the isosbestic point clearly reveals an approach to a reaction completion and easily corrects for this error (Figure 2c3). As this example nicely illustrates, normalization to an isosbestic point is ideally suited to correct for routine errors arising from changing total signal intensities, which are commonly encountered in continuous and discontinuous spectroscopic reaction monitoring. Collectively, the examples highlighted here may serve as case studies for how isometric points and the lack thereof can shine light on secondary reactions and correct for potential errors originating from these.
Finally, it is worth mentioning that there are scenarios where spectroscopic analysis meets its limits, with or without an isometric point. Although isometric points are immensely useful, not all transformations can be tuned to exhibit one under conditions which are compatible with the reaction system under investigation. For instance, reliable continuous monitoring of the phosphorolysis of 5 a by UV spectroscopy requires at least pH 10, which might not lie within the working space of the enzyme in question. In this case and in others like it, these types of issues can be resolved by uncoupling the reaction of interest from the spectral investigation. For instance, at pH 13, the anions 5 a À and 5 b À exhibit a stable isosbestic point. Thus, resorting to discontinuous monitoring by quenching samples of the 5 a!5 b transformation in strong base converts a messy set of spectra into a simplified set which can easily be analyzed by isometric normalization. [19] In cases where the presence of three or more observable compounds prevents the use of an isometric point, spectral deconvolution can help to draw useful information from messy spectra. [19][20][21][22][23] However, the success of deconvolution still hinges on the differentiability of the underlying spectra. If two or more spectra are too similar (i. e., they look too alike), deconvolution algorithms will struggle immensely to return useful data from highly overlapping spectra. Therefore, it is generally worth attempting to trace a messy system back to one that exhibits a clear isometric point as this tremendously increases the data quality available from experimental spectra.

Could You Summarize That Briefly?
While chemical transformations often yield convoluted and highly overlapping spectroscopic data, isometric points can help one draw useful information from such datasets. Isometric points are points in a spectrum with a constant signal intensity throughout the progress of a chemical transformation. An analysis with isometrically normalized signal intensities grants direct access to the degree of conversion of a transformation (through Equation (1)), if one has suitable reference spectra for the starting material and product. Thus, without any prior knowledge about the reactants, one can quantitatively follow their interconversion from messy spectroscopic data. through linear combination following: I y = x I y,A + (1À I y,B ) (2) where I y is the intensity at the point (here, a wavelength) y (arbitrary dimension), I y,A and I y,B are the intensities at the point y of A and B (the same arbitrary dimensions), and x is the molar fraction of A in the A-B mixture. The linear relationships drawn in Figure 1a were obtained through isometric normalization, using the isometric (here, the isosbestic) point ψ with the higher signal intensity (at 316) and the arbitrary analysis points 298 (ϕ 1 ) and 340 (ϕ 2 ). The values shown represent the ratios of the signal intensities at ϕ and ψ (where the intensity at ψ is already 1 in this case). One could draw a completely analogous graph using one of the other isometric points (at 252 or 266); this will only change the values of the isometrically normalized signal intensities I ϕ/ψ . It is generally advisable to use isometric points with higher signal intensities as this reduces the proportion of random noise present in that reference signal. Similarly, analysis points should be chosen as points where relevant change happens. Points near the baseline or very close to the isometric point show little change in signal intensity and will generally not yield meaningful data.
[10] These data were originally published in Figure S2e and S2 f of ref. [9].
The conditions used to obtain these data and the mathematical operations performed to yield the pK a value of 1 b are described in detail in the Supporting Information of the original publication. [12] These IR data as generated by Hutchinson, Welsh and Burés exhibit multiple isosbestic points. However, since the data are spaced in intervals of ca. 1.8 cm À 1 and both isosbestic points shown here lie squarely between two measured datapoints, accurate normalization is not possible here. However, since the data already nicely intercept at these isosbestic points, I regarded these raw data as normalized and treated them as such. Thus, I calculated conversion, using equation (1) with the intensity values at 0 min after catalyst addition (1.86 · 10 À 4 ) and 260 min after catalyst (À 0.00141) as references for E and P. From this conversion, I obtained concentration data through the mass balances of isovaleraldehyde (initial concentration of 10.3 mM) and 2 (18.5 mM). With isovaleraldehyde as the limiting reagent, 100 % conversion is equal to a conversion of 10.3 mM 2 to 2*. ). This yielded conversion data which could be approximated with a firstorder kinetic model of the form: c(t) = c eq À c eq exp(À kt) (3) where c(t) is the extent of conversion over time (dimensionless), t is the reaction time (minutes in this case) c eq is the extent of conversion in the equilibrium (dimensionless), and k is the rate constant (min À 1 ). This gave k = 0.061 � 0.002 min À 1 (R 2 = 0.998) with the data shown and c eq set to 100 %. [16] This is an interesting example. NMR spectra normally do not exhibit isointensic points. However, these do, so we can treat them as such. To derive the data shown in Figure 1e, I first subjected each spectrum to peak deconvolution with a Lorentzian function, using Origin 2022b's Multiple Peak Fit function. Using the fitted values for the areas corresponding to both peaks and the baseline, I corrected all spectra to a uniform baseline and normalized each spectrum by its total peak area. While the raw data do reflect the trends shown here quite well, the spectra derived via this procedure are quite a bit more illustrative. Indeed, calculating with the raw data would give quite similar results. To derive the conversion data shown Figure 1e, I normalized each spectrum by its intensity at the pseudo-isointensic point at 8.36887 ppm and evaluated their signal intensity at an analysis point of 8.36642 ppm (near the peak of the ADP signal). Using these isometrically normalized signal intensities and the molar distribution of ADP and ATP derived from the peak deconvolution, I calculated the linear relationship between the isometrically normalized signal intensity and the molar fraction of ADP by linear fitting. This yielded reference values of 4.587 for ADP and 0.992 (via 4.587-3.595) for ATP. Using these reference values, I calculated conversion through Equation (1) and approximated the resulting reaction course with a first order kinetic model following the relationship: c(t) = c eq À (c eq À c 0 ) exp(À kt) (4) where c(t) is the extent of conversion over time (dimensionless), t is the reaction time (minutes in this case) c eq is the extent of conversion in the equilibrium (dimensionless), c 0 is the extent of conversion prior to reaction initiation (dimensionless), and k is the rate constant (min À 1 ). This gave c 0 = (42.44 � 1.57)%, c eq = (70.37 � 0.85)%, and k = 0.51 � 0.06 min À 1 (R 2 = 0.988), when omitting the two datapoints shown in light gray. Although NMR data are -normally -clearly not meant to be treated this way, the results from this extrapolated analysis are in good agreement with the expected values and the results originally generated by Braham, Zalar, and Golovanov (ref. [15]).