Device and method for determining reaction kinetics

Abstract

A method of determining the activation energy E.sub.a for degradation of a chemical species includes in sequence the steps of a) simultaneously incubating a plurality of samples of the chemical species in a single unitary device at a plurality of constant temperatures T, in each case for an incubation time t selected to result in loss of at most 20 mol % of the amount originally present; b) quenching each of the samples to stop degradation; c) determining the mole fraction m of the chemical species remaining in each of the quenched samples, relative to the amount present before incubating; d) determining for each sample a reaction rate coefficient k.sub.obs according to the equation $k_{\mathrm{obs}}\left(T\right)=\frac{1-m\left(T\right)}{t};$
and e) performing numerical regression of the k.sub.obs values obtained in step d) and the corresponding temperatures T in K to derive the activation energy E.sub.a according to the following equation $k_{\mathrm{obs}}=k_0\exp \left(\frac{E_a}{R}\left(\frac{1}{T}-\frac{1}{T_0}\right)\right),$
or to derive a temperature-dependent activation energy if that is more appropriate for the chemical species of interest.

Claims

1. A method of determining the reaction rate coefficient (k.sub.obs) for the degradation of a chemical species at each of a plurality of constant temperatures, comprising in sequence the steps of a) simultaneously incubating a plurality of samples of the chemical species in a single unitary device at said plurality of constant temperatures T, wherein the incubation of each of the plurality of samples is performed for an incubation time t selected to result in loss of a portion of the chemical species, said portion being at most 20 mol % of the amount originally present, where the choice of t might or might not be the same for each value of T; b) quenching each of the samples in a manner sufficient to stop degradation; c) determining the mole fraction m of the chemical species remaining in each of the quenched samples, relative to the amount present before incubating; and d) determining for each sample a reaction rate coefficient k.sub.obs according to the equation $k_{\mathrm{obs}}\left(T\right)=\frac{1-m\left(T\right)}{t}.$

2. The method of claim 1, wherein the chemical species is a pharmaceutical product.

3. The method of claim 1, wherein the chemical species is a protein.

4. The method of claim 3, wherein the degradation comprises aggregation.

5. The method of claim 3, wherein the degradation comprises non-native aggregation.

6. The method of claim 3, wherein the degradation comprises chemical degradation.

7. The method of claim 1, wherein the loss of the chemical species is at most 10 mol %.

8. The method of claim 1, further comprising e) performing numerical regression of the k.sub.obs values obtained in step d) and the corresponding temperatures T in K to derive the activation energy E.sub.a of the degradation of the chemical species according to the following equation $k_{\mathrm{obs}}=k_0\exp \left(\frac{E_a}{R}\left(\frac{1}{T}-\frac{1}{T_0}\right)\right)$ wherein k.sub.0 is the value of k.sub.obs for T.sub.0, one of the plurality of temperatures in K.

9. The method of claim 8, wherein the chemical species is a pharmaceutical product.

10. The method of claim 8, wherein the chemical species is a protein.

11. The method of claim 10, wherein the degradation comprises aggregation.

12. The method of claim 10, wherein the degradation comprises non-native aggregation.

13. The method of claim 10, wherein the degradation comprises chemical degradation.

14. The method of claim 8, wherein the loss of the chemical species is at most 10 mol %.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) FIG. 1 is a schematic drawing of an exemplary device for performing the PTIR method according to the invention.

(2) FIG. 2 shows DSC traces for IgG1 (solid) and Fc-IgG1 (dashed) formulated in 10 mM acetate buffer at pH 4 (A), pH 5(B), and pH 6 (C) with no added salt or with 100 mM NaCl added salt (offset 150 kcal/mol). Curves are offset vertically for easier visualization.

(3) FIG. 3 shows exemplary isothermal and PTIR results for determining IgG1 aggregation rates at pH 5, (5 mM citrate buffer), and in the same buffer but with the addition of 100 mM NaCl. (A) Isothermal monomer loss versus time at 325.5 K (triangles), 330 K (circles), and 332 K (squares shown in the inset). (B) PTIR monomer loss at 2 hours (triangles, circles, diamonds, and open squares) or 24 hours (closed squares) as a function of incubation temperature. (C) Calculated ln(k.sub.obs) values derived from PTIR data.

(4) FIG. 4 shows an Arrhenius plot of the natural logarithm of the aggregation rate, or equivalently ln(k.sub.obs) as a function of inverse temperature, derived from PTIR data and showing comparison to results from traditional isothermal experiments. Closed symbols are from the PTIR data, and open symbols are from the traditional method. Different data sets are for different solution conditions, summarized below.

(5) FIG. 5 shows an Arrhenius plot using PTIR data for 2 hour and 24 hour incubations across a broader range of solution conditions when compared to FIG. 4.

(6) FIG. 6 shows effective aggregation activation energies for IgG1, determined from PTIR data as a function of pH and with or without 100 mM added NaCl. (A) 10 mM acetate (B) 5 mM citrate.

DETAILED DESCRIPTION OF THE INVENTION

(7) The inventors have now developed a Parallel-Temperature-Initial-Rates (PTIR) method that accurately and efficiently determines degradation rates as a function of temperature. An exemplary apparatus for doing this is also provided.

(8) In conventional approaches, one determines monomer loss for multiple samples at a small number of temperatures (sometimes only one, e.g., 40 C.) over a predetermined incubation time (e.g., according to ICH guidelines). In the PTIR method of this invention, one instead determines monomer loss for a small number of samples (e.g., one sample in the extreme example below) at multiple temperatures for the same incubation time. That is, rather than take measurements at multiple incubation times at a given temperature, one takes measurements at multiple temperatures for a given incubation time. The incubation time is chosen to result in loss of at most 20 mol % of the protein monomer, or at most 10 mol %. The amount of monomer loss need only be enough to allow accurate and precise measurement. Typically, it will be at least 0.5 mol %, or at least 1, 2, or 4 mol %.

(9) The aggregation rate for a given temperature from PTIR agrees with results from a traditional, isothermal method. However, PTIR has the advantage of providing activation energy (E.sub.a) values that are either impractical to obtain with any reasonable certainty using traditional approaches, or are fundamentally of limited accuracy if one uses temperature-scanning methods. Within this context, it should be noted that E.sub.a values from experimental data necessarily have statistical uncertainty or confidence intervals (i.e., error bars) because they require one to regress data versus a model such as the Arrhenius equation. Traditional isothermal methods require many samples at each temperature, and therefore have practical limits in that E.sub.a values are determined using a small number of temperatures. This necessarily leads to large uncertainties in E.sub.a values, making the data of limited use for setting product shelf lives or quantifying the stability of products as a function of temperature. The PTIR method provides accurate initial rate values for a given temperature, as well as activation energies having much better precision and reliability (i.e., small error bars) than those obtained by conventional approaches. For example, the error bars for the E.sub.a values from the data by the conventional approach in examples below are as large as the E.sub.a values themselvesthis makes them statistically meaningless for prediction of sample stability at lower temperatures. In contrast, the PTIR based E.sub.a values have error bars that are much smaller and provide much more reliable interpolation and extrapolation of aggregation rates to other temperatures of interest. See Table 1 below. The PTIR method can easily be adapted for use with degradation routes other than non-native aggregation, and provides an efficient method to more accurately determine the temperature dependence of protein degradation rates, and potentially to predict long-term protein stability at much lower temperatures outside the range measured by a user with the PTIR method.

(10) In some embodiments of the invention, the PTIR method provides rate coefficient (units of inverse time), k.sub.obs, for protein aggregation processes, as well as activation energies E.sub.a for such processes. It also provides k.sub.obs and E.sub.a values for total degradation of proteins, i.e., losses due to aggregation and/or chemical degradation. For example, k.sub.obs and E.sub.a values for chemical degradation losses of proteins due to hydrolysis, for example deamidation, can be determined by this approach. More generally, these values can be determined for chemical species in general, for example drugs or pharmaceutical products of any type.

EXAMPLES

General Experimental Procedures

(11) AS-IgG1 (>98% monomer) was provided by Amgen as a stock solution at a concentration of 30 mg/mL. Additionally, purified fragment crystallizable region (Fc-IgG1) was provided by Amgen as a stock solution at a concentration of 20 mg/ml. The protein was dialyzed as previously reported by Brummitt et al. J. Pharm. Sci. 100 (2011) 2087-2103 and by Kim et al., Biophys. Chem. 172 (2013) 26-36. The protein concentration was confirmed using UV-Vis absorbance at 280 nm (Agilent 8453 UV-Vis, Agilent Technologies, Santa Clara, Calif.) using an IgG1 extinction coefficient of 1.586 mL/mg cm and an Fc-IgG1 extinction coefficient of 1.36 mL/mg cm. All solutions were diluted gravimetrically to working concentrations.

(12) Monomer concentrations were quantified using size exclusion chromatography (SEC). An Agilent 1100 HPLC (Agilent Technologies, Santa Clara, Calif.) was connected in-line to a Tosoh (Montgomeryville, Pa.) TSK-Gel 3000xL column. Samples were injected with an autosampler (100 microliter injections), with samples held at room temperature prior to injection. Concentration was determined by peak area, using a variable wavelength detector (Agilent technologies, Santa Clara, Calif.) and absorbance at 280 nm, with external standards. Additional details are the same as previously reported by Kim et al.

(13) Differential scanning calorimetry (DSC) was performed using standard techniques as previously reported by Brummitt et al. and Kim et al.

(14) Incubations were performed as follows. IgG1 stock solutions were prepared at 1 mg/mL at a given pH, NaCl concentration, and buffer type, and aliquotted into hermetically sealed deactivated borosilicate glass HPLC vials (Waters, Milford, Mass.). Isothermal incubations were performed by heating multiple samples at a given temperature in a water bath, or in the custom-built PTIR device shown schematically in FIG. 1, and removing samples at pre-determined incubation times. Incubation temperatures were chosen such that samples could easily be removed at multiple points during the early periods of monomer loss, i.e., when m=1 to 0.8, where m is defined as the concentration of monomer divided by the initial monomer concentration, as measured by SEC peak area. At each time point, a given vial was immediately quenched by immersion in an ice-water bath to arrest aggregation, and was subsequently held at room temperature (20-23 C.) prior to analysis with SEC.

(15) FIG. 1 is a schematic drawing of an exemplary device shown generally at 10, suitable for performing the PTIR method according to the invention. The device has a lid 12, an insulation plate 14 having placement windows 16 through which vials 26 can be placed into wells 24 in heating blocks 22, which are made of a thermally conductive metal, for example aluminum. The temperature of each of the heating blocks 22 is controlled independently by its own Peltier device (not shown), assisted by contact of the heating blocks 22 with a metal heat sink 18 whose temperature is maintained by flow of a heating or cooling liquid through passages 20. For ease of reference, the heating blocks 22 and the metal heat sink 18 are shown in FIG. 1 in a raised position relative to housing 28, but in practice they are enclosed within it. Housing 28 sits on base 32, and contains the Peltier devices and their electrical connections (not shown), which interface with one or more controller units (not shown) via electrical connectors 30.

(16) In the particular embodiment shown in FIG. 1, the device has 10 independent Peltier-controlled metal heating blocks each capable of holding two 1.5 mL HPLC vials. The Peltier-controlled blocks are cooled with circulating water to maintain stable temperature control, and are insulated from each other and the surroundings.

(17) In the present Examples, incubation temperatures were chosen so that samples quenched after 2 or 24 hours would have monomer loss values that fell in the initial-rate regime, i.e., at most 20 mol % loss of monomer. The incubation time was selected to be not less than 2 hours so as to allow for sufficient temperature equilibration and elimination of artificial lag times at shorter incubation time scales when samples were heating to the set-point temperature. Sample temperatures were confirmed independently with a separately calibrated thermocouple. The longer incubation timescale of 24 hours was chosen to achieve initial rates approximately one order of magnitude slower than 2 hours. For some examples, 10-day incubations were also performed.

(18) Calculation of Aggregation Rates by PTIR Method

(19) Aggregation rates were determined from the monomer fraction remaining as a function of incubation time, measured by SEC as described above. Over approximately the first ten to twenty percent monomer loss, the rate of change of m remains nearly constant and the observed rate law can therefore be well described as zeroth order without the need to assume an underlying rate law. Therefore, in the traditional method, for each temperature the monomer fraction was regressed with Equation [1] to obtain the aggregation rate coefficient (units of inverse time), k.sub.obs, from the regime where m was between approximately 1 and 0.8, i.e., loss of at most 20 mol % of the protein monomer.
m=1k.sub.obst[1]

(20) Aggregation rates, in units of 1/time, in the initial-rate regime are equivalent to rate coefficients, and are determined using the PTIR approach are based on Equation [2], which is derived by rearranging Eq. 1 and solving for k.sub.obs.

(21) $\begin{array}{cc}{k}_{\mathrm{obs}}\left(T\right)=\frac{1-m\left(T\right)}{t}& \left[2\right]\end{array}$
In Eq. 2, temperature (T) is the variable of interest when using the PTIR method, as t is held constant for a given experiment. The PTIR analysis method is valid for initial-rate conditions, where the rate of degradation remains approximately constant. This is also expected to hold for other degradation processes (e.g., chemical degradation with breaking of covalent bonds) not investigated here, as the principle of initial rates in reaction kinetics is more general than just the example shown here.
Differential Scanning Calorimetry (DSC) to Guide the Choice of Incubation Temperatures

(22) If one has preliminary data to indicate the rate of aggregation at one temperature, the following is unnecessary. But if one is starting with no knowledge of aggregation rates for the protein and solution conditions of interest, the following way is useful for determining a starting point for PTIR in terms of guiding which temperature range to use for measuring aggregation rates.

(23) FIG. 2 shows thermograms for AS-IgG1 (solid) and the corresponding Fc fragment (dashed) at pH 4 (panel A), pH 5 (panel B), and pH 6 (panel C) in 10 mM acetate. Previous work reported DSC thermograms for AS-IgG1 at the same pH and NaCl concentrations, but in 5 mM citrate buffer. In FIG. 2, profiles for conditions with 100 mM added NaCl are offset vertically to distinguish them from those with no added NaCl. The peaks of the Fc-IgG1 thermograms overlay with the smaller peaks or shoulders of the full IgG1 thermogram in panels A to C. As expected based on previous literature reports, there are only two, relatively small, transitions for the Fc-IgG1 when compared with the full IgG1; the peak at lower (higher) temperature is assigned to the C.sub.H2 (C.sub.H3) domain of the Fc. For the full IgG1, the peak for the Fab domains overlaps with one or both peaks from the Fc domains, depending on the solution pH. The DSC profiles are consistent with the pH dependent thermograms reported in the literature for a range of other IgG1 molecules as a function of pH and salt concentration. Prior work by Kim et al. showed that aggregation involved unfolding of the Fab domain for this IgG1.

(24) Aggregation Rates from PTIR Versus Standard Isothermal Approaches

(25) The DSC thermograms were used to guide the initial choices for incubation temperatures for accelerated aggregation rates. All incubation temperatures were selected to be below the DSC Fab peak temperatures for a given solution condition, based on the discussion above. Aggregation rates were determined using the PTIR method and quantitatively compared to those determined by canonical isothermal-rate experiments. Aggregation rates or initial-rate coefficient (k.sub.obs) values were calculated based on Eq. 1 or 2 above. Using a single sample at each temperature for the PTIR approach provides a worst case example, as one could easily use more than one incubation time or replicate samples at a given incubation time for each temperature. However, the results below indicate that this may not be necessary if one has sufficiently high-precision results with the assay of choice (e.g., SEC in the present case).

(26) FIG. 3 compares the results obtained by standard isothermal monomer loss kinetics (panel A) with those obtained by the PTIR method (panel B), using 1 mg/mL IgG1 in pH 5 buffer (5 mM citrate) with 100 mM added NaCl. Panel A shows isothermal monomer loss as a function of incubation time (t) for 325.5 K (triangles), 330 K (circles), and 332 K (squares shown in the inset). Visual inspection of FIG. 3, panel A reveals that monomer loss was linear versus t over the experimental range tested (m=1-0.8). Previously reported work also showed linear kinetics during initial periods of aggregation, which is expected based on general mass action kinetic considerations when the extent of reactant consumption is small. When operating in the initial-rate regime, one does not need to know or assume the reaction mechanism. However, if one operates at much larger extents of monomer loss (m<<1), the monomer loss profile is expected to become non-linear, and this requires one to determine or assume the underlying aggregation mechanism to accurately quantify the net or observed rate coefficient for monomer loss (k.sub.obs).

(27) FIG. 3 panel B shows aggregation rates determined by using the PTIR method according to the invention. The symbols show m as a function of incubation temperature for 2-hour (circles, diamonds, triangles and open squares) and 24-hour (closed squares) incubation times. Different temperatures were used for the 2-hour and for the 24-hour experiments, based on the discussions above and below. For the 2-hour experiments, three separate protein stocks were prepared and the experiment was repeated on separate days to provide a simple assessment of variability. Scatter in the data in FIG. 3 panels B and C shows typical experimental uncertainty expected with the PTIR approach. The value of m for each symbol in FIG. 3 panel B was converted to k.sub.obs using Eq. 2, with the corresponding values of ln(k.sub.obs) being given in FIG. 3 panel C. The symbols in panel C correspond to those in panel B for a given solution condition, for a given set of temperature values. Error bars are smaller than the size of the symbols unless visible in either panel.

(28) As the PTIR approach is valid in the initial rate regime (m=1 to approx. 0.8), aggregation rates measured at longer times necessarily correspond to incubations at lower temperature(s). The 24-hour time-scale experiments were chosen to extend the range of accessible k.sub.obs values by at least an order of magnitude. Ten-day incubations were also performed (data not shown in FIG. 3), but in many cases significant fragmentation occurred, and this convolutes the interpretation and analysis to properly determine monomer loss rates for monoclonal antibodies. No data are reported here, or were used in subsequent analysis, for which fragmentation was evidenced in SEC.

(29) The results in FIG. 3 illustrate that one obtains temperature-dependent rates of aggregation for many temperatures (e.g., 15 values in FIG. 3 panel B) with the PTIR method, while using the same amount of protein material as was required to obtain rates for only three temperatures via standard methods (i.e., FIG. 3 panel A). The overlapping data sets in FIG. 3 panel B are from repeat experiments using freshly prepared protein stock solutions on separate days, to illustrate that the PTIR results are robust and reproducible. This example illustrates the PTIR method, its reproducibility, and its comparable material needs to standard methods for determining rates at a single or small set of temperatures. The examples below show that the PTIR method provides accurate rate values when compared to the standard method for a single temperature, and superior results when considering the temperature-dependence of aggregation rates.

(30) Comparison of PTIR and Isothermal Incubations

(31) FIG. 3 illustrates the standard method (panel A) and the PTIR method (panel B) for a single solution condition. FIG. 4 summarizes results analogous to those in FIG. 3 for AS-IgG1 for a range of solution conditions, and also compares the rates determined from PTIR (across a range of temperatures) with those from the standard method applied at particular temperatures. The results in FIG. 4 compare the values of the logarithm of k.sub.obs versus inverse temperature (i.e., an Arrhenius diagram) for the different methods, demonstrating that the PTIR approach allows one to efficiently and accurately measure temperature-dependent aggregation rates across a range of temperatures. Each data set corresponds to a different solution condition with 5 mM sodium citrate buffer: pH 5, no added NaCl (diamonds); pH 5, 100 mM added NaCl (triangles); pH 6, no added NaCl (squares); and pH 6, 100 mM added NaCl (closed circles). The open symbols in FIG. 4 show the k.sub.obs values determined from traditional isothermal incubation (i.e., analogous to the data in FIG. 3 panel A), while closed symbols are for the PTIR method (i.e., analogous to the data in FIG. 3 panel B). The conditions for FIG. 3, panels A and B are the same as those for the closed diamonds in FIG. 4.

(32) The aggregation rates determined via the PTIR approach are comparable in accuracy to those obtained by the standard isothermal aggregation method, which uses many samples at the same temperature, while the PTIR method provides rates for many more temperatures, using comparable consumption of protein material and user time. As noted above, the results provided here are a worst case example for accuracy using PTIR, in that only a single incubation time was used for a given temperature in the PTIR method.

(33) Determination of Activation Energies

(34) An important use for values of k.sub.obs vs. T is to determine an accurate value of the effective activation energy (E.sub.a) of aggregation, so that accelerated aggregation rates may be more effectively extrapolated to lower temperaturese.g., for predicting room temperature shelf life. Intuitively, having k.sub.obs values at more T values will allow one to regress E.sub.a values with much better statistical confidence intervals and will provide greater ability to extrapolate k.sub.obs to lower temperatures.

(35) FIG. 4 shows an Arrhenius diagram of aggregation rates determined using the PTIR device and traditional isothermal incubations. The results show so-called Arrhenius behavior, in that the data are effectively linear when plotted as ln k.sub.obs vs. 1/T. Based on FIG. 3, it is apparent that if one uses comparable amounts of material for both PTIR and traditional isothermal experiments, PTIR provides almost an order of magnitude increase in the number of temperatures that can be tested. This increased information regarding the temperature dependence of the rates provides for much more reliable E.sub.a values, as illustrated below.

(36) The data were regressed using the Arrhenius equation (Eq. 3), where k.sub.obs is the experimentally determined value for reaction rate coefficient (units of inverse time), E.sub.a is defined above, k.sub.0 is the value of k.sub.obs at an arbitrarily chosen temperature, T.sub.0. In each case below, k.sub.0 was a fitting parameter and T.sub.0 was selected as 333.15 K because that is near the median of all incubations temperatures. Choosing different values for T.sub.0 shifted the fitted value and confidence interval for k.sub.0, but not for E.sub.a Values of T are in K.

(37) $\begin{array}{cc}{k}_{\mathrm{obs}}=k_0\exp \left(\frac{E_a}{R}\left(\frac{1}{T}-\frac{1}{T_0}\right)\right)& \left[3\right]\end{array}$

(38) Table 1 reports the pre-factors (k.sub.0) and activation energies (E.sub.a) from Arrhenius fits using data plotted in FIG. 4 for PTIR and canonical isothermal incubations. Three temperatures were used for the canonical method (see open symbols in FIG. 4). Notably, current guidelines from regulatory agencies for accelerated stability tests for pharmaceutical products require even fewer than three incubation temperatures.

(39) TABLE-US-00001 TABLE 1 PTIR Method Canonical Isothermal Method k.sub.0 (hr.sup.1) E.sub.a (kcal/mol) k.sub.0 (hr.sup.1) E.sub.a (kcal/mol) pH 5 0 mM NaCl (9.5 0.9) 10.sup.3 156 12 (7 20) 10.sup.3 200 300 pH 5 100 mM NaCl 0.122 0.015 123 6 0.12 0.12 140 50 pH 6 0 mM NaCl (2.3 0.8) 10.sup.4 220 20 (0.5 5) 10.sup.3 200 400 pH 6 100 mM NaCl (1.2 0.2) 10.sup.3 206 15 (1.4 3.8) 10.sup.3 200 200

(40) Visual inspection of Table 1 shows the PTIR approach results in much smaller confidence intervals for fitted parameters. In contrast, the confidence intervals on the parameters regressed for the standard isotherm experiments are so large that the fitted parameters are statistically meaningless (e.g., the uncertainty in E.sub.a is as large as the value of E.sub.a itself). These results further illustrate the effectiveness of using the PTIR method and device to determine temperature-dependent aggregation rates, as well as illustrating greatly improved determination of activation energies if one chooses to regress rate data using the Arrhenius equation.

(41) Shelf-Life Prediction and Non-Arrhenius Rates

(42) Although not shown herein, the PTIR approach can be extended to longer incubation times (multiple days to months) to yield results that may be predictive of rates at even lower temperatures (e.g., refrigerated or room-temperature conditions). This would be straightforward for any degradation route for which Eq. 3 is an accurate description. For example, deamidation and, more generally, hydrolysis reactions in aqueous solution, are typically Arrhenius over the relevant temperature range for pharmaceutical products.

(43) In the present case, aggregation rates were measured over relatively small temperature windows (i.e., net change in rates on the order of 10.sup.2), and therefore an Arrhenius equation is expected to be valid. If one does not need to extrapolate more than approximately one order of magnitude (e.g., a factor of 10-20, or from 1 month to almost 2 years), it is reasonable to expect that the Arrhenius equation will hold reasonably well for such an extrapolation if the data under accelerated conditions show Arrhenius behavior (Roberts et al. Int. J. Pharm. 48:318-333 (2011)). As the desired shelf life for typical pharmaceutical products is at least 18 months, and usually is targeted for 18-24 month, the present invention can greatly improve the ability of workers to make reliable predictions of product shelf life.

(44) Alternatively, it has been reported that, when considering rates over a much broader range of time scales (i.e., many orders of magnitude), non-Arrhenius behavior may become significant for protein aggregation. The term non-Arrhenius in this context refers to cases where the activation energy depends on temperature (i.e., data are curved or show a kink or breakpoint in a diagram of the type shown in FIG. 4). In this case, one is advised to not rely on Eq. 3, or its analogous mathematical forms, for use with the entire temperature range. Rather, one should either regress k.sub.obs vs. temperature to a non-Arrhenius model if one has confidence that model is appropriate, or one should use only the PTIR data from the temperatures closest to those of interest for shelf life predictions. That is, rather than regress a model such as the Arrhenius model to data spanning many orders of magnitude for k.sub.obs, one instead focuses on determining the local activation energy based on the k.sub.obs data that span one to two orders of magnitude and lie at the temperatures closest to those of interest for shelf life predictions. The idea of using a local E.sub.a values is equivalent to using the local slope, or tangent, to a curve in numerical integration. Such an idea was suggested by one of the inventors previously (Wang and Roberts, AAPS J. (online only) doi: 10.1208/s12248-013-9485-3 (2013)), but this approach requires accurate rate data for many temperatures. As such, the PTIR method enables shelf life prediction in such non-Arrhenius situations.

(45) Discriminating Effects of pH, Buffer Type, and NaCl on Aggregation Rates and E.sub.a Values

(46) Knowledge of activation energies is important for extrapolating rates and predicting shelf lives, but also has value for improved understanding of the underlying degradation rates, as this may influence decisions regarding which conditions will be optimal for a given product. The present example extended the case above for aggregation of AS-IgG1 to compare the buffer type (acetate vs. citrate) for the same pH and NaCl ranges used above.

(47) Values of k.sub.obs(T) were determined as a function of pH (4, 5, 6), added NaCl concentration (0 mM or 100 mM), and buffer species (citrate or acetate). FIG. 5 is an Arrhenius diagram based on the PTIR method for all solution conditions that were tested, including 2 hour and 24 hour incubations. Symbols shown in FIG. 5 are as follows.

(48) TABLE-US-00002 Shape pH mM NaCl Buffer Triangle, open 6 100 5 mM citrate Triangle, closed 4 100 5 mM citrate Triangle, split 5 100 5 mM citrate Circle, open 6 0 5 mM citrate Circle, closed 4 0 5 mM citrate Circle, split 5 0 5 mM citrate Hexagon, open 5 100 10 mM acetate Hexagon, closed 5 0 10 mM acetate Star, open 6 100 10 mM acetate Star, closed 6 0 10 mM acetate Diamond, open 4 100 10 mM acetate

(49) Previously reported work qualitatively showed that changing buffer species could significantly alter aggregation rates. Additionally, the results above indicate that IgG1 formulated at pH 4 in 10 mM acetate buffer with no added salt resulted in no aggregation (monomer loss data not shown) even after heating at 85 C. for one hour.

(50) From visual inspection of FIG. 5, it is clear that no single incubation temperature would be practical to achieve aggregation rates on a comparable timescale (hours to weeks) for all solution conditions. For example, if one selected a temperature to achieve a rate corresponding to ln(k.sub.obs)=7 for solution conditions indicated with the closed circles, the rates for solution conditions depicted with closed triangles would be so large as to be impractical to measure (and vertically far off-scale in FIG. 5). This highlights another utility of the PTIR approach, i.e., that it allows one to obtain k.sub.obs(T) profiles for head-to-head comparison between solution conditions or different proteins that would otherwise be untenable to quantitatively compare if one had to select a common temperature for measuring aggregation rates.

(51) FIG. 6 shows E.sub.a values and 95% confidence intervals determined from fitting 2-hr PTIR data to Eq. 3 for each of the formulation conditions in FIG. 5, with the panels A and B for those formulations prepared in 10 mM acetate buffer, and 5 mM citrate buffer, respectively. For each pH, formulations prepared without added NaCl are shown as open bars and those with 100 mM NaCl concentration are shown as closed bars. Notably, pH 4 with no added NaCl and 10 mM acetate had an unmeasurably large E.sub.a value, as IgG1 heated in this formulation did not aggregate. The question mark in the open bar at pH 4 in FIG. 6 (panel A) indicates that E.sub.a is unknown for that condition because aggregation was too slow to measure.

(52) At 100 mM added NaCl, E.sub.a increases with increasing pH, which is expected as conformational stability (i.e., T.sub.m.sup.app) increases with pH. Based on previously reported thermodynamic arguments and qualitative mechanistic arguments for non-native aggregation, larger T.sub.m.sup.app values imply increased unfolding enthalpy values, and therefore higher E.sub.a values. However, E.sub.a values at low ionic strength conditions (no added NaCl) for acetate buffer show the opposite behavior. That is, T.sub.m.sup.app values decrease as one decreases pH, but E.sub.a values increase substantially.

(53) The present results highlight that both conformational stability and inter-protein interactions can play a discernable role in determining aggregation rates (monomer loss). Although the canonical isothermal method was not employed in this example, the results in Table 1 and FIG. 4 illustrate that one can expect error bars or uncertainties in E.sub.a values from traditional methods that are almost a factor of 10, or larger, compared to those from the PTIR method when using comparable amounts of material. If one were to increase the error bars to that extent in FIG. 6, the important trends and differences would not be discernable within statistical uncertainty. As such, PTIR provides a means to determine more accurate and reliable activation energies that can be enabling for a variety of applications, including more rational and accurate selection of optimal product formulation conditions.

(54) Although the invention is illustrated and described herein with reference to specific embodiments, the invention is not intended to be limited to the details shown. Rather, various modifications may be made in the details within the scope and range of equivalents of the claims and without departing from the invention.