Sigrid Pieters

Y Vender Heyden

M. d Jean-Michel M Roger

' Hondt

Laurent Hansen

Bernard Palagos

Bart De Spiegeleer

Jean-Paul P Remon

Chris Vervaet

Thomas De Beer

Yvan Vander Heyden

Matthias D'hondt

Raman spectroscopy and multivariate analysis for the rapid discrimination between native

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

The FDA's process analytical technology (PAT) initiative [1] is one of the concepts encouraging the (bio-) pharmaceutical industry to invest in analytical methods that provide ultra-fast (i.e. real-time) analysis results, while being information-rich and robust in the presence of interferents. Most of them rely on spectroscopic techniques combined with multivariate analysis (MVA) [START_REF] Beer | Nearinfrared and Raman spectroscopy for the in-process monitoring of pharmaceutical production processes[END_REF][START_REF] Dingari | Raman spectroscopy provides a powerful diagnostic tool for accurate determination of albumin glycation[END_REF][START_REF] Li | Rapid characterization and quality control of complex cell culture media solutions using Raman spectroscopy and chemometrics[END_REF]. As they usually lack the conventional delay inherent to the standard analytical techniques for sampling, sample preparation, analysis, data extraction and interpretation, not only the laboratory work, time and costs may be reduced, but they may also allow decision making in real-time. This is essential for using these techniques as in-situ diagnostic tools during manufacturing processes, when the quality attribute of interest has to be assessed within a few seconds to make important decisions that ensure overall product and process quality.

The 3-D folded structure of a native protein is required to properly exert its biological action. The most important bonds that make up the secondary to quaternary structure are hydrophobic interactions (e.g. between the side chains of the amino acids) and hydrogen bonds (e.g. for constituting the secondary structural elements such as α-helix and β-sheets).

However, these structures are fragile, since exposure to even mildly denaturating conditions may make them lose partially their natively-folded structure. These conditions can be numerous, and freeze-drying is well known to be one of them [START_REF] Hill | Thermodynamic and dynamic factors involved in the stability of native protein structure in amorphous solids in relation to levels of hydration[END_REF][START_REF] Kasper | The freezing step in lyophilization: physico-chemical fundamentals, freezing methods and consequences on process performance and quality attributes of biopharmaceuticals[END_REF][START_REF] Carpenter | Rational design of stable lyophilized protein formulations: some practical advice[END_REF]. When the protein does not refold back to its native state after reconstitution, its activity might be reduced or absent.

Partial unfolding can also result in the exposure of buried hydrophobic amino acid residues. This may favour aggregation when reconstituting the freeze-dried product with typically an up-concentrated protein fraction [START_REF] Niazi | Handbook of biogeneric therapeutic proteins: regulatory, manufacturing, testing, and patent issues[END_REF]. To avoid excessive unfolding and subsequent exposure of the hydrophobic sites it is thus necessary to take preventive actions, conserving the native conformation of the proteins as much as possible during the whole manufacturing process and

Author-produced version of the article published in European Journal of Pharmaceutics and Biopharmaceutics, 2013, N°85(2), p. The original publication is available at http://www.sciencedirect.com/science/article/pii/S0939641113001264 Doi: 10.1016/j.ejpb.2013.03.035 storage. The potential risk for reduced or absent activity of the non-native proteins, and immune responses of the aggregates, makes them important considerations in formulation and process development of protein drug products.

Among the many available techniques that may provide information on a protein's 3Dstructural conformation (e.g. fluorescence spectroscopy, far-UV circular dichroism, NMR, FTIR...), Raman spectroscopy may have the potential to become an ultra-fast diagnostic tool in detecting in real-time protein conformational changes during the freeze-drying process.

Whereas other techniques require the protein to be in solution, and/or are invasive (i.e. FTIR would affect the freeze-drying process) Raman spectroscopy is a non-invasive technique which is practically transparent for water and ice, and requires minimal or no sample preparation. Near-infrared (NIR) spectroscopy can also detect protein unfolding during freeze-drying, however due to its strong water and ice signals it could only be used for monitoring a sample after sublimation in the illuminated sample spot [START_REF] Pieters | Near-infrared spectroscopy for in-line monitoring of protein unfolding and its interactions with lyoprotectants during freeze-drying[END_REF]. Infrared (IR) and Raman spectroscopy have been extensively used in protein structural characterization, especially for determining changes in the secondary and/or tertiary structure of proteins in different matrices. Identification of specific bands associated with the secondary structure has been achieved using homopolypeptides and proteins with known α-helix and βsheet elements [START_REF] Barth | What vibrations tell us about proteins[END_REF]. As these methods require a thorough peak-by-peak investigation to obtain the necessary information from the spectra of the protein formulations, they usually demand blank corrections, resolution enhancement techniques and expert knowledge. Visual interpretation of the amide I FTIR spectral band (1700-1600 cm -1 ) is by far the most established technique for determining secondary structure of proteins in the solid state [START_REF] Barth | What vibrations tell us about proteins[END_REF].

As the original amide I spectrum is usually featureless and can be overlapped by the signals of other components, the visualization of the underlying secondary structure components is hampered. Therefore, prior to visual inspection blank corrections and decomposition methods, such as Fourier deconvolution or second derivation of the spectra, have become popular to delineate overlapping secondary structures in the original spectrum [START_REF] Barth | What vibrations tell us about proteins[END_REF]. As delineating is more complex in the blank corrected Raman spectra, the interpretation is frequently based on changes in peak shape rather than on individual band assignments [START_REF] Mangialardo | Raman analysis of insulin denaturation induced by high-pressure and thermal treatments[END_REF][START_REF] Elkordy | Study of protein conformational stability and integrity using calorimetry and FT-Raman spectroscopy correlated with enzymatic activity[END_REF][START_REF] Tuma | Raman spectroscopy of proteins: from peptides to large assemblies[END_REF]. Usually, the focus is on the visual interpretation of the well-known amide I and amide III bands.

Although aforementioned methods are very informative, they have the drawback of being time consuming and unpractical for a rapid quality assessment, which is essential for inline monitoring of the protein conformational state during freeze-drying processes. The proposed MVA method uses the Raman spectrum more efficiently, by extracting the relevant information directly from the sample spectrum, without the need for blank corrections, resolution enhancement or subjective interpretations. Multivariate calibration methods can cope with non-selectivity in the obtained spectral signals, provided the interferent signals are sufficiently incorporated into the calibration and their shape is not completely identical to the signal of the analyte [START_REF] Bro | Multivariate calibration: What is in chemometrics for the analytical chemist?[END_REF]. Additionally, MVA may actually point out these spectroscopic marker bands differentiating between native and non-native proteins, thus offering insights into the biochemical origin of the discrimination. Mathematical approaches may also allow detecting more subtle, but consistent, spectral changes that are visually undetectable [START_REF] Dingari | Raman spectroscopy provides a powerful diagnostic tool for accurate determination of albumin glycation[END_REF].

In the present paper we investigate whether Raman spectroscopy combined with MVA enables the discrimination between native-like and non-native conformational states of proteins in freeze-dried formulations in a rapid (i.e. a few seconds) and direct way. The aims are: (1) to investigate whether Raman spectroscopy is able to rapidly and accurately differentiate between native-like and non-native freeze-dried proteins, (2) to develop and validate a calibration model to classify freeze-dried protein samples, (3) to study the influence of formulation interferents (i.e. excipient signals) on the discrimination, (4) to examine which part(s) of the Raman spectra are necessary to make this discrimination, and whether they can

Author-produced version of the article published in European Journal of Pharmaceutics and Biopharmaceutics, 2013, N°85(2), p. The original publication is available at http://www.sciencedirect.com/science/article/pii/S0939641113001264 Doi: 10.1016/j.ejpb.2013.03.035 be linked to sound biochemical knowledge. Given its known instability during freeze-drying, lactate dehydrogenase (LDH) was selected as a model protein [START_REF] Al-Hussein | The effect of mannitol crystallization in mannitol-sucrose systems on LDH stability during freeze-drying[END_REF].

Methods and materials

Materials

L-Lactic Dehydrogenase (LDH) from rabbit muscle -Type II in ammonium sulfate was obtained from Sigma-Aldrich (Saint Louis, MO, USA). Prior to freeze-drying, 2 mL type I vials (Nipro, Authon-du-Perche, France) were filled with 720 µl formulation containing approximately 9 mg/ml LDH. Table 1 shows the different LDH formulations that were prepared in this study. Formulations 1-12 were used in the design of experiments (DoE) and contained a 40 mM buffer pH 7.4, either sodium phosphate (Sigma Aldrich, Steinheim, Germany) (PHOS), L-histidine (HIS) or TRIS (Merck, Darmstadt, Germany). Some formulations contained a lyoprotectant, i.e. 5% sucrose (Suedzucker, Mannheim, Germany) and/or a cryoprotectant, i.e. 1% PEG 4000 (Fagron, Barsbüttel, Germany). Besides the DoE formulations, others (formulations 13-19) contained NaSCN, NaCl or ethanol (Sigma Aldrich, Steinheim, Germany), or were formulated in water.

Freeze-drying

Freeze-drying was performed using an Amsco FINN-AQUA GT4 freeze-dryer (GEA, Koeln, Germany). Eight batches were produced with different cooling methods and secondary drying temperatures (Table 2). The cooling methods included: (1) slow cooling at a rate of 0.2°C/min (SL), (2) fast cooling with pre-cooled shelf at -45°C (FA), (3) cooling at 1°C/min and after freezing an annealing step of 10h at -7°C (ANN), (4) cooling at 1°C/min with cooling stop for 60 min at -10°C (CH). Freezing was always performed for 2h at -45°C, primary drying for 20h at -20°C and 100 µbar, and secondary drying for 6h at 100 µbar with temperature ramps of 0.2°C/min in between.

Author-produced version of the article published in European Journal of Pharmaceutics and Biopharmaceutics, 2013, N°85(2), p. The original publication is available at http://www.sciencedirect.com/science/article/pii/S0939641113001264 Doi: 10.1016/j.ejpb.2013.03.035

The full factorial experimental design led to 96 freeze-dried samples (8 runs x 12 formulations) (Tables 12). Seven additional samples (formulations [START_REF] Tuma | Raman spectroscopy of proteins: from peptides to large assemblies[END_REF][START_REF] Bro | Multivariate calibration: What is in chemometrics for the analytical chemist?[END_REF][START_REF] Al-Hussein | The effect of mannitol crystallization in mannitol-sucrose systems on LDH stability during freeze-drying[END_REF][16][START_REF] Vandeginste | Handbook of chemometrics and qualimetrics: Part B[END_REF][START_REF] Fisher | The use of multiple measurement in taxonomic problems[END_REF][START_REF] Barker | Partial least squares for discrimination[END_REF] were added to runs 1 and 2 to produce more extremely denatured LDH. This resulted in a total of 110 freezedried samples that were analyzed with both FTIR and Raman spectroscopy. After reconstituting the samples, they were analyzed using the LDH activity assay. However, since the amide I bands in the FTIR spectra of some freeze-dried samples could not be interpreted, those were omitted and the data set for developing and validating the model comprised 99 freeze-dried samples.

LDH activity assay

The enzymatic activity of LDH after reconstitution of the freeze-dried samples in 720 µl water was determined with a UV-2401PC spectrophotometer (Shimadzu, Kyoto, Japan) by monitoring the rate of absorbance decrease per minute of NADH at 340 nm as a result of the reduction of pyruvate to lactate at 37°C. The Sigma-Aldrich (Steinheim, Germany) experimental procedure was followed [16], using all their necessary chemicals (i.e. NADH, sodium pyruvate and BSA). The remaining activity of the reconstituted freeze-dried samples was compared to that of native (not freeze-dried) LDH, and expressed as % LDH activity recovery after freeze-drying. Each sample was measured in triplicate and averaged. The assay had a coefficient of variation of 4.1% for repeatability and 11.0% for day-to-day intermediate precision. nm. Spectra were recorded with a resolution of 4 cm -1 and an exposure time of 40 s, using a laser power of 400 mW. For MVA, the Raman spectra of the freeze-dried solids were used without any pre-processing. For visual comparison of the Raman spectra, the corresponding blank spectrum of each formulation was subtracted and the spectra were scaled and Savitzky-Golay smoothed.

Spectroscopic conditions and spectral

FTIR spectroscopy

FTIR spectroscopy functioned as a reference technique to evaluate potential changes in protein secondary structure in the freeze-dried solid. FTIR spectroscopy measurements were conducted on a Varian Scimitar 800 FTIR spectrometer, equipped with a Specac Golden Date diamond ATR module (Varian Scimitar, Middelburg, The Netherlands). The spectra were recorded from 4000 to 500 cm -1 in ATR mode at a resolution of 2 cm -1 . Each measurement was the average of 100 scans. The FTIR spectra of the same freeze-dried samples and the corresponding blanks were first background (air) corrected and converted to absorbance format. The resulting FTIR absorbance spectra were subsequently corrected for the spectral contributions of the blank formulations. Finally, the second derivative with 17-point Savitzky-Golay smoothing was performed and the FTIR spectra were area normalized over the amide I region (1700-1600 cm-1) for direct comparison. Individual peak positions were identified and assigned according to the literature [START_REF] Barth | What vibrations tell us about proteins[END_REF].

Data handling and MVA

All data handling was performed with Matlab 7.1 (The Mathworks, Natick, MA) and the PLS toolbox 6.2. (Eigenvector Research, Wenatchee, WA). For PLS-LDA and orthogonal projections in-house Matlab routines were used. Statistical data analysis was performed with Minitab 16 (Minitab, PA).

Principal Component Analysis (PCA)

Author-produced version of the article published in European Journal of Pharmaceutics and Biopharmaceutics, 2013, N°85(2), p. The original publication is available at http://www.sciencedirect.com/science/article/pii/S0939641113001264 Doi: 10.1016/j.ejpb.2013.03.035

Principal Component Analysis (PCA) is an unsupervised MVA technique that allows seeing trends and structures in data sets. It reduces the number of variables to a new set of latent variables (LV's) that describe the data variability more efficiently. These LV's, called principal components, are linear combinations of the original ones. They are defined in such a way that they explain most the (remaining) variability in the data and are orthogonal to each other. They can be decomposed into loadings and scores. The loading vector gives qualitative information about the samples by describing what type of information characterizes them, i.e.

which variables are important. The associated weighted averages of the original variables are the scores, providing quantitative information, i.e. the amount of each loading vector in each sample [START_REF] Bro | Multivariate calibration: What is in chemometrics for the analytical chemist?[END_REF][START_REF] Vandeginste | Handbook of chemometrics and qualimetrics: Part B[END_REF].

Partial Least Squares -Linear Discriminant Analysis (PLS-LDA)

PLS-LDA is a supervised classification algorithm that combines PLS for dimension reduction with LDA for obtaining a classification outcome. LDA seeks an optimal lowdimensional space in which the projected data points are linearly well-separated into different classes [START_REF] Fisher | The use of multiple measurement in taxonomic problems[END_REF]. It aims to maximize the ratio of between-class variance to within-class variance.

The Mahalonobis distance, taking the shape of the considered cluster into account, is used to calculate the probability that a new object belongs to a certain class. As the number of variables (i.e. wavenumbers) in the Raman and FTIR spectral data is much higher than the number of objects (i.e. samples), LDA cannot directly be applied to the spectra. Therefore, PLS first compresses the data dimensionality [START_REF] Barker | Partial least squares for discrimination[END_REF]. PLS is performed on a set of dummy responses (0 and 1) encoded to represent the 2 classes, and maximizes the co-variance between the measured signals (e.g. spectra) and the response [START_REF] Vandeginste | Handbook of chemometrics and qualimetrics: Part B[END_REF].

For building the PLS-LDA models, the data set of 99 samples was randomly split into a training (75 freeze-dried LDH formulations) and a test set (24 freeze-dried LDH Author-produced version of the article published in European Journal of Pharmaceutics and Biopharmaceutics, 2013, N°85(2), p. The original publication is available at http://www.sciencedirect.com/science/article/pii/S0939641113001264 Doi: 10.1016/j.ejpb.2013.03.035 formulations for validation of the model), but it was made sure that all possible variation in y (i.e. containing native-like, mildly stressed (i.e. mainly altered ratio in α-helix and β-sheet structures, further called α/β ratio) and more strongly stressed (i.e. with the clear presence of non-native structures) LDH with % activity preservations ranging from <10% to >100%) was covered by both data sets. Both training and test set contained formulations with similar excipients originating from the 8 freeze-drying runs. The optimal model complexity was determined with cross validation (CV) on the training set. A permutation test was also part of the validation procedure [START_REF] Westerhuis | Assessment of PLSDA cross validation[END_REF]. This can assess whether the classification of the samples in the two designed groups performs significantly better than any other random classification in two arbitrary groups. This test should exclude the possibility of chance correlation in the model.

For every permutation test, the class labels were randomly assigned to different samples. The permutation test was repeated a hundred times to obtain a distribution of classification expected to be non-significant.

Orthogonal projections to study the influence of the excipients

Protein formulations usually require excipients, such as buffers (here histidine, phosphate or TRIS), lyo-and/or cryo-protectants (here sucrose and PEG), because of their delicate physical stability. As we directly use the unprocessed Raman spectra of a freeze-dried formulation (without the usual blank subtraction), we have to investigate the influence of the excipient signals on the discriminating power of the model. Therefore we applied an orthogonal projection approach [START_REF] Skibsted | Net analyte signal based statistical quality control[END_REF][START_REF] Hansen | Detecting variation in ultrafiltrated milk permeates -infrared spectroscopy signatures and external factor orthogonalization[END_REF][START_REF] Pinto | Advantages of orthogonal inspection in chemometrics[END_REF][START_REF] Boulet | Pretreatments by means of orthogonal projections[END_REF]. The contributions within the variable space of the calibration matrix X (containing the spectra of the protein formulations) are originating from the pure protein signals (X p ), interferences such as the excipient signals (X b ) and other undefined spectral variance (ε) (eq. 1).

Author-produced version of the article published in European Journal of Pharmaceutics and Biopharmaceutics, 2013, N°85(2), p. The original publication is available at http://www.sciencedirect.com/science/article/pii/S0939641113001264 Doi: 10.1016/j.ejpb.2013.03.035

ε + + = b p X X X
(1)

In practice, the interference space X b was estimated via an external data set containing only the Raman spectra of the freeze-dried blanks (B) of all possible formulations used in the data set (Table 1 and Fig. 1) [START_REF] Skibsted | Net analyte signal based statistical quality control[END_REF]. The column vectors of the so obtained matrix B form a basis of the detrimental subspace. To remove those variations in X originating purely from excipients signals, an orthogonal projector to B, i.e. ⊥ B P , can be calculated. This corrects X as follows (eq. 2) to obtain X*.

) ) ( ( * T 1 T B B B B I X X X - ⊥ - = = B P ( 2
)
where I is the identity matrix.

Thus, the algorithm finds the dimensions of the spectral space which are affected by the studied excipients and projects the spectra orthogonal to this variation. After the orthogonalization procedure, it can be assessed whether the discrimination and the classification between freeze-dried formulations with native-like and non-native proteins is affected, this way revealing the influence of the removed factor [START_REF] Hansen | Detecting variation in ultrafiltrated milk permeates -infrared spectroscopy signatures and external factor orthogonalization[END_REF].

Results and discussion

Generation of a relevant data set

Whether and to which extent a protein will lose its native conformation depends upon the environmental conditions it is exposed to, and can be very erratic. Therefore, a relevant data set to study whether there can be discriminated between native-like and non-native states of proteins in freeze-dried formulations should cover sufficient native-like samples, as well as a distribution of denatured types (i.e. from mildly to more strongly stressed) obtained by freeze-drying. Therefore, the freezing method, secondary drying temperature and formulation 12). An ANOVA was performed on the % LDH activity recovery of the 96 DoE samples, to investigate the influence of 5 process and formulation parameters on the preservation of the native protein conformation after reconstitution of the freeze-dried product (Table 3). Three factors, i.e. freezing method, buffer type and sucrose addition, showed statistically significant effects (α=0.05) on the % LDH activity recovery. Slow cooling showed a negative effect (i.e. it showed the lowest activities) on the % LDH activity recovery compared to the other tested cooling methods. The use of phosphate buffer showed a negative effect compared to the others, and the addition of 5% sucrose as a protectant to the formulation positively affected the LDH activity preservation.

FTIR as a reference technique to assign class labels

Since FTIR easily allows analysis in the solid freeze-dried state, is well-known and frequently used, and is complementary to Raman spectroscopy, this technique was chosen as a reference for obtaining information on the protein conformation in the freeze-dried samples.

To assign class membership labels (i.e. NL and DEN) to the samples, a two-step approach was used. First, we visually compared the experimental versus the native reference amide I FTIR spectrum to assign each sample to a class. Next, we performed a PCA on the spectra of the complete labeled sample set to independently confirm the correctness of the initial class labeling.

Changes in the distribution of secondary structures of freeze-dried proteins are most commonly assessed by visual comparison of the second derivative of the amide I band (1700-1600 cm -1 ) in the blank corrected FTIR spectra of the sample with a native reference spectrum [START_REF] Barth | What vibrations tell us about proteins[END_REF][START_REF] Al-Hussein | The effect of mannitol crystallization in mannitol-sucrose systems on LDH stability during freeze-drying[END_REF][START_REF] Pelton | Spectroscopic methods for analysis of protein secondary structure[END_REF][START_REF] Haris | FTIR spectroscopic characterization of protein structure in acqueous and non-acqueous media[END_REF][START_REF] Kong | Fourier transform infrared spectroscopic analysis of protein secondary structures[END_REF][START_REF] Dong | Infrared spectroscopic studies on lyophilization-and temperature-induced protein aggregation[END_REF][START_REF] Izutsu | Protection of protein secondary structure by saccharides of different molecular weights during freeze-drying[END_REF][START_REF] Jackson | The use and misuse of FTIR spectroscopy in the determination of protein structure[END_REF]. Freeze-dried formulations were designated as NL if the amide I spectrum was similar (in secondary structural elements distribution) to that of the native LDH before freeze-drying (Fig. 2). The amide I region of native LDH indicated primarily alpha helix structures (near 1658 cm -1 ), to a lesser extent beta-sheets (near 1636 cm -1 ), and also a very small portion of intermolecular beta-sheets (near 1620 cm -1 ), which is consistent with the literature [START_REF] Al-Hussein | The effect of mannitol crystallization in mannitol-sucrose systems on LDH stability during freeze-drying[END_REF]. The small frequency and intensity shifts of the main bands in their freeze-dried versus liquid form can be related to the altered hydration state of the protein molecules [START_REF] Barth | What vibrations tell us about proteins[END_REF][START_REF] Walsh | The hydration of amides in helices: a comprehensive picture from molecular dynamics, IR, and NMR[END_REF]. Since it was not possible to preserve the native state in the freeze-dried product without Fig. 3B. Signals of non-native structures appeared in the amide I FTIR spectra around 1660-1680 cm -1 (turns and different unordered structures) and around 1620 cm -1 (intermolecular βsheets). Protein unfolding was also indicated by band broadening in these regions, where clear absorbance signals at particular wavenumbers are lost due to a fading of structural homogeneity among the protein population [START_REF] Allison | Effects of drying methods and additives on structure and function of Actin: mechanisms of dehydration-induced damage and its inhibition[END_REF]. These samples will provide most likely the highest quality risk at the time of reconstitution, since the refolding of the protein to its native state may be more hampered or impossible [START_REF] Niazi | Handbook of biogeneric therapeutic proteins: regulatory, manufacturing, testing, and patent issues[END_REF]. This was also seen by their generally lower % LDH activity recovery after reconstitution.

The data set of 99 samples that was used for the exploratory analysis, model development and model validation, comprised 31 NL and 68 DEN samples. Overall, the 99 freeze-dried formulations comprised a wide distribution of % LDH activity recoveries (Fig. 4). It can be seen that the reconstituted NL samples always had high % LDH activity recoveries (ranging from 88.1% to 109.3%), while those of DEN samples varied between 0.4% and 109.6%. The latter is the consequence of having different kinds of protein denaturation in the freeze-dried state, i.e. reversible and irreversible denaturation upon reconstitution. Both NL and DEN groups contained formulations with similar excipients (except samples formulated with NaSCN, NaCl, ethanol and pure water that were always DEN), originating from the 8 freeze-drying runs.

To independently confirm the visual assignment of the class labels to the samples, a PCA was performed on the full sample set constituting the same labeled amide I FTIR spectra as used for the visual interpretation. As spectral changes due to altered protein conformation largely exist in this set of samples, this type of systematic variation must be represented in the primary PC's. The separation among the scores of the NL and DEN samples in the PC1-PC2 scores plot (Fig. 5) was an independent justification for the correctness of the class labelling.

Raman spectroscopy: Exploratory analysis and classification

To explore trends in the Raman spectra related to the conformational state of the LDH, PCA was performed on the Raman spectra of the data set of 99 samples. No pre-processing of the spectra was applied since different methods (i.e. standard normal variates, normalization, multiplicative scatter correction, Savitsky-Golay smoothing, derivatives) did not improve the discrimination performance (data not shown). Different parts of the spectra within the 800-1800 cm -1 range were assessed, since the range below 800 cm -1 is known to be less sensitive for protein conformation [START_REF] Herrero | Raman spectroscopy for monitoring protein structure in muscle food systems[END_REF]. The PC2-PC6 score plot of the Raman spectra in the range 1033-1725 cm -1 showed for the majority of the samples a discrimination between both NL and DEN states, but the scores of 5 DEN samples could not be differentiated from the NL group (Fig. 6).

The PLS-LDA model built from Raman spectra required 6 LV's and showed 5% (4 samples) misclassifications in CV, while 1 sample of the 24 test set samples was classified wrongly. The discrimination performance of the PLS-LDA model before and after orthogonal projection of the matrix of blank spectra (B) to the 75 spectra of the calibration matrix (X) was unchanged (Table 4), which indicates that the model is able to predict the protein's conformational state in the presence of this variation, while not being influenced by this variation. Hence, the signals of the tested excipients are not a confounding factor towards discrimination, neither positively nor negatively. Fig. 7 also shows that one is still able to discriminate for most of the samples between both NL and DEN states in a PCA score plot.

This finding is of importance from a practical point of view. It implies that all the necessary information can be extracted from a single Raman spectrum of the sample, and that no corresponding blanks (at least for the tested excipients) should be freeze-dried nor measured, while a blank correction is required for visual interpretation of Raman (and FTIR) spectra. The wrongly classified samples were always mildly denatured LDH (see their amide I FTIR spectra in Fig. 3A). These samples were assigned as DEN by the reference technique, but were predicted by the model as NL samples. In these samples, it was mainly the α/β ratio that was slightly changed, and almost no non-native structures were formed in the freezedried product. Their remaining LDH activities after reconstitution ranged between 90.3% and 108.9%. This is similar with those from NL samples, which implies that the denaturation is reversible upon reconstitution. These results suggest that the Raman model has lower sensitivity in detecting mildly denatured freeze-dried samples than the reference technique. It is indeed expected that when there is a continuum of states from NL to various kinds of DEN there will be a limit for detecting first signs of protein denaturation. These results suggest that this limit is higher for the Raman model than for the visual interpretation of the FTIR spectra.

When comparing the number of misclassifications for the Raman model (in green) to the predictions of the models generated by the random permutations of the class labels, we noticed a clear distinction between the permutation distribution and the original classification (Fig. 8). When comparing the original classification with all the permutations a p value of <0.001 could be obtained at the 0.01 significance level, both for CV and self prediction. This confirms that the classification is not based on spurious correlations but rather on consistent differences in spectral features between NL and DEN samples. It was studied whether these correlations could be linked to biochemical knowledge.

Spectral regions responsible for discrimination

From Fig. 9 it was seen that the loading vector of PC6 and the discriminant vector of the PLS-LDA model had very similar shapes. PC6 described only 0.03% of the variation in the spectra, which means that there is a lot of other variation in the Raman spectra, not responsible for the discrimination between both classes (e.g. light scattering, physical and environmental effects, instrumental effects...) [START_REF] Pinto | Advantages of orthogonal inspection in chemometrics[END_REF][START_REF] Kjeldahl | Some common misunderstandings in chemometrics[END_REF]. Yet the loading vector of PC6 or the discriminant vector can be informative for studying the most important variables responsible for the discrimination. They indicated that the NL and DEN samples have maximum differences in the following regions of the Raman spectrum: (i) near 1070 cm -1 (C-N stretch),

(ii) near 1130 cm -1 (NH 3 deformation), (iii) 1200-1320 cm -1 (amide III), and (iv) 1320-1500 cm -1 (mainly CH n non-stretching with possible participations of CN and CC stretching).

Surprisingly, the amide I band (1600-1700 cm -1 ) was only slightly indicated. As the indicated bands comprised both backbone (i.e. amide III) and side chain sensitive regions, we tried to link them with biochemical knowledge. Therefore, we also visually inspected the blank corrected Raman spectra and the corresponding amide I FTIR spectra of some relevant NL and DEN samples (Fig. 10). When visually comparing the Raman spectra of those samples, we see that major spectral differences between NL and DEN samples are manifested in similar spectral regions as indicated by the loading and discriminant vectors. To study more in depth the influence of the different regions (linked to certain biochemical knowledge) of the Raman spectrum, different PLS-LDA models were built using each of these spectral regions separately or combined.

Backbone sensitive spectral regions

The amide I (peak with maximum near 1665 cm -1 ) and III (1200-1320 cm -1 ) Raman bands are well known to be related to the protein's backbone conformations (secondary structural elements) [START_REF] Herrero | Elucidation of structural changes in soy protein isolate upon heating by Raman spectroscopy[END_REF][START_REF] Maiti | Raman spectroscopic characterization of secondary structure in natively unfolded proteins: αsynuclein[END_REF]. Their shapes are the result of overlapping components corresponding to αhelices, β-sheets, turns and random structures. Therefore broadening of the peak width and/or differences in shape of the amide I and III bands suggest an altered distribution of secondary structures (native and/or non-native) [START_REF] Herrero | Elucidation of structural changes in soy protein isolate upon heating by Raman spectroscopy[END_REF][START_REF] Maiti | Raman spectroscopic characterization of secondary structure in natively unfolded proteins: αsynuclein[END_REF][START_REF] Hédoux | In situ monitoring of proteins during lyophilization using micro-raman spectroscopy: A description of structural changes induced by deshydration[END_REF][START_REF] Maiti | Raman spectroscopic characterization of secondary structure in natively unfolded proteins: α-Synuclein[END_REF]. However, among the PLS-LDA models built from different separate amide ranges (Table 5), the ones using the amide I and II bands showed a very high number of misclassifications, both for CV and for test set prediction. It confirms the knowledge that the amide II Raman band has little sensitivity for protein secondary structure determination [START_REF] Herrero | Elucidation of structural changes in soy protein isolate upon heating by Raman spectroscopy[END_REF][START_REF] Tantipolphan | Insights into the structure of proteins by vibrational spectroscopy[END_REF]. This was also seen in Fig. 10A when visually inspecting the amide II band. Despite its very well known sensitivity for protein secondary structures, the poor performance of the PLS-LDA model built from the amide I band indicates that this band is little suitable for differentiation between NL and DEN in the studied formulations. This is also noticed when visually inspecting the amide I bands in the blank corrected Raman spectra (Fig. 10A). The amide I band of a DEN sample in a sucrose containing formulation resembles more that of a NL sample with sucrose than that of a DEN sample without sucrose. It is well known that co-lyophilized mixtures of sucrose and polymers (e.g. proteins) interact at a molecular level through hydrogen bonding, forming miscible amorphous molecular dispersions [START_REF] Taylor | Sugar-polymer hydrogen bond interactions in lyophilized amorphous mixtures[END_REF][START_REF] Randolph | Phase separation of excipients during lyophilization: effects on protein stability[END_REF][START_REF] Allison | Hydrogen bonding between sugar and protein is responsible for inhibition of dehydration induced protein unfolding[END_REF]. As there is evidence that the C=O stretching vibration is strongly influenced by the polymer-sucrose hydrogen bond [START_REF] Taylor | Sugar-polymer hydrogen bond interactions in lyophilized amorphous mixtures[END_REF], it is expected that this will notably influence the amide I band, typically dominated by signals of the backbone C=O stretching vibrations. Thus the strong influence of the extent of hydrogen bonding of the protein with the environment (i.e. hydration state) on the amide I Raman band is here possibly the reason for the modest discrimination between NL and DEN when only considering this band. Among the tested backbone sensitive regions, the amide III band clearly showed being the most suitable for building the model.

Side chain sensitive spectral regions

In Fig. 10A there were also differences between NL and DEN samples in the spectral regions related to some amino acid side chains, i.e. the bands 1320-1500 cm -1 , and near 1130 cm -1 and 1070 cm -1 [START_REF] Herrero | Elucidation of structural changes in soy protein isolate upon heating by Raman spectroscopy[END_REF][START_REF] Ostrovskii | Raman spectroscopic characterization of association and thermoreversible gelation in aqueous systems of poly(N-acetamidoacrylamide)[END_REF]. This matches again with the variables indicated by the loading and discriminant vectors. The performances of the PLS-LDA models built from these separate side chain sensitive regions are shown in Table 5, and could further improve when combining the different regions. Major changes in the CH n non-stretching region (1320-1500 cm -1 ) were linked to a change in 'exposure' of the hydrophobic amino acid residues to the outer surface, i.e. buried versus exposed to the environment, as they were likewise assigned for heatdenatured proteins [START_REF] Herrero | Elucidation of structural changes in soy protein isolate upon heating by Raman spectroscopy[END_REF]. This can be considered as an indication of protein unfolding, since hydrophobic residues will be buried in a natively folded protein, yet a larger fraction of them will be exposed in the partially unfolded states. Wang et al. [START_REF] Wang | Optofluidic device for ultrasensitive detection of proteins using surface-enhanced Raman spectroscopy[END_REF] also found drastic changes in the 1050-1200 cm-1 region of heat-denatured insulin and attributed this to protein unfolding.

Combined spectral regions

The classification performance of the PLS-LDA models built from fused backbone or side chain sensitive spectral regions surpassed those from single isolated bands (Table 5). Overall, the models considering the fused side chain sensitive regions and the full range (1033-1725 cm -1 ) (Table 4) showed the highest accuracy. The latter model was preferred because it combines information from complementary sources, i.e. backbone (i.e. amide III) and side chain sensitive regions.

Conclusions

This on samples from novel batches is still required. This feasibility study suggests positive outlooks for Raman spectroscopy for the direct and automated analysis of the investigated quality attribute. However, as one analytical method can never fully characterize the conformational state of proteins, the proposed method rather aims giving a quality indication.

The speed and automatic nature of the method advocate that a transfer of the Raman model to in-line and real-time diagnostic use during freeze-drying processes may be a possible future step. The Raman spectra were the result of the difference between the sample and corresponding blank spectra, and were subsequently Savtizky-Golay smoothed and normalized. The FTIR spectra were pre-processed as described in Methods and Materials. The black dotted line in (B) corresponds to the average spectrum of 5 NL freeze-dried LDH formulations.

Author-produced version of the article published in European Journal of Pharmaceutics and Biopharmaceutics, 2013, N°85(2), p. The original publication is available at http://www.sciencedirect.com/science/article/pii/S0939641113001264 Doi: 10.1016/j.ejpb.2013.03.035

  sucrose, the NL formulations contained sucrose. Membership to the NL class was defined by two criteria: (1) no clear changes in the distribution of the native secondary structural elements, and (2) no clear formation of non-native structures, compared to the native state spectrum.Freeze-dried samples showing signs of not preserving the fully native structure were designated as DEN. As protein denaturation in the freeze-dried product was diverse, some examples of the amide I FTIR spectra of DEN proteins are shown in Fig.3. They are compared to the average of 5 native-like freeze-dried LDH formulations (bold dotted black line). Non-native LDH in the freeze-dried state was the result of an altered α/β ratio (i.e. clear changes in the peak intensities near 1658 and 1636 cm -1 ), or a decrease of these structures with the formation of non-native structures (i.e. clearly appearing peaks near 1620 cm -1 and in the region 1660-1680 cm -1 )[START_REF] Barth | What vibrations tell us about proteins[END_REF]. The samples where the former phenomenon was mainly present can be seen as mildly stressed since they still comprise largely of native secondary structures. Some examples are shown in Fig.3A. The remaining LDH activity after reconstitution of these samples ranged between 90.3 and 108.9%, which may reflect their stronger tendency to refold back to the native conformation during reconstitution. Hence, they may provide a lower quality risk. Some of them had sucrose in their formulation, which probably helped preventing more excessive unfolding. Several DEN freeze-dried samples also showed clearly the presence of non-native structures in their amide I FTIR spectrum. Some examples with various % LDH activity recoveries in their reconstituted product are shown in

Fig. 1 .

 1 Fig. 1. Raman spectra of the blanks of the investigated formulations.

Fig. 2 .

 2 Fig. 2. Second derivative amide I FTIR spectrum of native LDH (in liquid) before freeze-

Fig. 3 .Fig. 4 .

 34 Fig. 3. Second derivative spectra of the amide I band (FTIR) of freeze-dried LDH

Fig. 5 .Fig. 7 . 5 Fig. 8 .

 5758 Fig. 5. PCA score plot (PC1-PC2) of the second derivative amide I FTIR spectra of the data

Fig. 9 .Fig. 10 .

 910 Fig. 9. Unprocessed Raman spectra of the data set (99 samples), the loading vector of the PC6

  Author-produced version of the article published in European Journal of Pharmaceutics and Biopharmaceutics, 2013, N°85(2), p. The original publication is available at http://www.sciencedirect.com/science/article/pii/S0939641113001264 Doi: 10.1016/j.ejpb.2013.03.035

  Author-produced version of the article published in European Journal of Pharmaceutics and Biopharmaceutics, 2013, N°85(2), p. The original publication is available at http://www.sciencedirect.com/science/article/pii/S0939641113001264 Doi: 10.1016/j.ejpb.2013.03.035 NL) and non-native or denatured (further called DEN) LDH in freeze-dried formulations (Tables

composition were varied to yield different relevant examples of native-like (further called

  Author-produced version of the article published in European Journal of Pharmaceutics and Biopharmaceutics, 2013, N°85(2), p. The original publication is available at http://www.sciencedirect.com/science/article/pii/S0939641113001264 Doi: 10.1016/j.ejpb.2013.03.035

  Author-produced version of the article published in European Journal of Pharmaceutics and Biopharmaceutics, 2013, N°85(2), p. The original publication is available at http://www.sciencedirect.com/science/article/pii/S0939641113001264 Doi: 10.1016/j.ejpb.2013.03.035

  Author-produced version of the article published in European Journal of Pharmaceutics and Biopharmaceutics, 2013, N°85(2), p. The original publication is available at http://www.sciencedirect.com/science/article/pii/S0939641113001264 Doi: 10.1016/j.ejpb.2013.03.035

  Author-produced version of the article published in European Journal of Pharmaceutics and Biopharmaceutics, 2013, N°85(2), p. The original publication is available at http://www.sciencedirect.com/science/article/pii/S0939641113001264 Doi: 10.1016/j.ejpb.2013.03.035

  Author-produced version of the article published in European Journal of Pharmaceutics and Biopharmaceutics, 2013, N°85(2), p. The original publication is available at http://www.sciencedirect.com/science/article/pii/S0939641113001264 Doi: 10.1016/j.ejpb.2013.03.035

  Author-produced version of the article published in European Journal of Pharmaceutics and Biopharmaceutics, 2013, N°85(2), p. The original publication is available at http://www.sciencedirect.com/science/article/pii/S0939641113001264 Doi: 10.1016/j.ejpb.2013.03.035

  Author-produced version of the article published in European Journal of Pharmaceutics and Biopharmaceutics, 2013, N°85(2), p. The original publication is available at http://www.sciencedirect.com/science/article/pii/S0939641113001264 Doi: 10.1016/j.ejpb.2013.03.035

  study, performed on a data set containing NL and a wide distribution of DEN states, demonstrated that Raman spectroscopy is with 95% accuracy able to rapidly and directly discriminate between NL and DEN states of LDH in various freeze-dried formulations. What is an acceptable criterion for this limit considering the intended (i.e. future in-line) use of the model? Also, further testing of the model performance Author-produced version of the article published in European Journal of Pharmaceutics and Biopharmaceutics, 2013, N°85(2), p. The original publication is available at http://www.sciencedirect.com/science/article/pii/S0939641113001264 Doi: 10.1016/j.ejpb.2013.03.035

	Misclassifications were always the result of assigning mildly (and reversible) DEN samples
	(with minimal changes in secondary structure detected by the reference technique) to the NL
	class. Hence, the sensitivity to detect minimal signs of protein denaturation is lower than the
	chosen reference approach. Therefore, the sensitivity of the Raman model should be further
	investigated with appropriate data sets. For instance, further research should include (1) What
	is the limit for detecting first signs of protein denaturation by the Raman model?, (2) How can
	sensitivity be improved? and (3)

Table 1 .

 1 Different LDH formulations (appr. 9 mg/ml LDH) prepared in this study.

	Formulation number	Excipient composition
	1	HIS
	2	HIS, sucrose
	3	HIS, PEG
	4	HIS, sucrose, PEG
	5	PHOS
	6	PHOS, sucrose
	7	PHOS, PEG
	8	PHOS, sucrose, PEG
	9	TRIS
	10	TRIS, sucrose
	11	TRIS, PEG
	12	TRIS, sucrose, PEG
	13	50 mM NaSCN
	14	150 mM NaSCN
	15	300 mM NaSCN
	16	600 mM NaSCN
	17	0.9% NaCl
	18	1.0% ethanol
	19	water
	Abbreviations: HIS= histidine buffer, PHOS = phosphate buffer, TRIS = TRIS buffer, all 40
	mM at pH 7.4.

Table 2 .

 2 Design of the freeze-drying runs performed.

	Batch Freezing	Secondary drying	Formulations*
		method	temperature (°C)
	1	SL	40	1-19
	2	ANN	40	1-19
	3	CH	5	1-12
	4	ANN	5	1-12
	5	FA	40	1-12
	6	FA	5	1-12
	7	CH	40	1-12
	8	SL	5	1-12
	Abbreviations: SL = slow cooling, FA = fast cooling, ANN = annealing, CH = cooling hold
	* See

table 1 .

 1 Author-produced version of the article published in European Journal of Pharmaceutics and Biopharmaceutics, 2013, N°85(2), p. The original publication is available at http://www.sciencedirect.com/science/article/pii/S0939641113001264 Doi: 10.1016/j.ejpb.2013.03.035

Table 3 .

 3 Factor level space of the studied process and formulation variables, their mean response per factor level and their significance on the % LDH activity preservation.

	Factor	Unit Levels Mean response	p value
	Freezing method (A)	-	SL	78.89	0.003*
			FA	88.94
			ANN	90.16
			CH	89.28
	Secondary drying temperature (B)	°C	5	85.19	0.174
			40	88.44
	Buffer type (C)	-	HIS	91.34	<0.001*
			PHOS	79.63
			TRIS	89.48
	Sucrose addition (D)	%	0	78.24	<0.001*
			5	95.40
	PEG addition (E)	%	0	86.68	0.908
			1	86.96
	*significant effect (α = 0.05).
	Abbreviations: SL = slow cooling, FA = fast cooling, ANN = annealing, CH = cooling hold,
	HIS= histidine buffer, PHOS = phosphate buffer, TRIS = TRIS buffer

Table 4 .

 4 Performance of the PLS-LDA models built from Raman spectra. Author-produced version of the article published in European Journal of Pharmaceutics and Biopharmaceutics, 2013, N°85(2), p. The original publication is available at http://www.sciencedirect.com/science/article/pii/S0939641113001264 Doi: 10.1016/j.ejpb.2013.03.035

	Technique	Spectral	Number of	% misclassifications	% misclassifications
		region	PLS	in CV	in test set
		(cm -1 )	components
	Raman	1033-1725 6	5	1/24
	Raman blank	1033-1725 4	5	1/24
	corrected*
	*after orthogonal projection to the blank matrix

Table 5 .

 5 Performance of the PLS-LDA models built from smaller spectral (or fused) regions of the Raman spectra assigned to certain protein information.Author-produced version of the article published in European Journal of Pharmaceutics and Biopharmaceutics, 2013, N°85(2), p. The original publication is available at http://www.sciencedirect.com/science/article/pii/S0939641113001264 Doi: 10.1016/j.ejpb.2013.03.035

	Spectral		Number of	%	%
	region		PLS	misclassifications	misclassifications
			components	in CV	in test set
	Band	Wavenumber
	assignment	(cm -1 )
	Amide I	1600-1700	5	33	7/24
	Amide II	1500-1600	8	23	12/24
	Amide III	1200-1320	6	7	2/24
	Backbone sensitive regions	6	5	2/24
	fused*
	CH n non-stretch 1320-1500	5	7	2/24
	C-N stretch +	1033-1200	4	5	3/24
	NH 3
	deformation
	Side chain sensitive regions	6	5	1/24
	fused**
	* fusion of the amide I, II, III regions
	** fusion of the CH n non-stretch, C-N stretch and NH 3 deformation regions

  Author-produced version of the article published in European Journal of Pharmaceutics and Biopharmaceutics, 2013, N°85(2), p. The original publication is available at http://www.sciencedirect.com/science/article/pii/S0939641113001264 Doi: 10.1016/j.ejpb.2013.03.035

	second derivative absorbance (log 1/R)	1700 -0.04 -0.03 -0.02 -0.01 0 0.01 0.02	1690	1680	1670	1650 decrease in 1660 alpha helix content	1630 increase in 1640 beta sheet content	1620	1610	1600
						wavenumber (cm-1)
		0.02
	second derivative absorbance (log 1/R)	-0.04 -0.03 -0.02 -0.01 0 0.01	turns		unordered		100.3% 28.6% 0.4% 45.0% 67.7% intermolecular beta-sheet beta sheet alpha helix
		1700 -0.05	1690	1680	1670	1660	1650	1640	1630	1620	1610	1600
						wavenumber(cm-1)

  Author-produced version of the article published in European Journal of Pharmaceutics and Biopharmaceutics, 2013, N°85(2), p. The original publication is available at http://www.sciencedirect.com/science/article/pii/S0939641113001264 Doi: 10.1016/j.ejpb.2013.03.035

		0.1
		0.08	DEN
	Scores on PC 2 (27.03%)	-0.06 -0.04 -0.02 0 0.02 0.04 0.06	NL
		-0.08
		-0.1
		-0.1	-0.08	-0.06	-0.04	-0.02	0	0.02	0.04	0.06	0.08	0.1
						Scores on PC 1 (32.72%)

  Author-produced version of the article published in European Journal of Pharmaceutics and Biopharmaceutics, 2013, N°85(2), p. The original publication is available at http://www.sciencedirect.com/science/article/pii/S0939641113001264 Doi: 10.1016/j.ejpb.2013.03.035

		2	x 10	4
									Raman spectra
	intensity	1 1.5
			1100 C-N stretch/ NH3 deformation 1200	1300	1400 non-stretching CHn 1500	1600	1700	0.05
							Raman shift (cm-1)	loading PC6
									0
						amide III		amide II	amide I
	0.02 0.04		1100	1200		1300	1400	1500	1600 discriminant vector 1700	-0.05
		0
	-0.02
	-0.06 -0.04		1100	1200	1300 Raman shift (cm-1) 1400	1500	1600	1700

Acknowledgements

SP is funded as an aspirant of the Research Foundation-Flanders (FWO). She would also like to thank the same institution and Irstea for the funding and hosting, respectively, of a research stay. MD and LH thank the Agency for Innovation by Science and Technology (IWT) for financial support.


Guido Bacciagaluppi
email: g.bacciagaluppi@abdn.ac.uk

Insolubility from No-Signalling

, showing that within the framework of the unitary Schrödinger equation it is impossible to reproduce the phenomenological description of quantum mechanical measurements (in particular the collapse of the state of the measured system) by assuming a suitable mixed initial state of the apparatus. The result follows directly from the nosignalling theorem applied to the entangled state of measured system and ancilla. As opposed to other 'insolubility theorems' for the measurement problem of quantum mechanics, it focuses on the impossibility of reproducing the phenomenological collapse of the state of the measured system.

Phenomenologically, a quantum measurement consists of a transformation of the state of a quantum system (the 'collapse' of the state) upon interaction with a suitable measuring apparatus. In each experimental setup ('measurement'), one of a family of transformations can take place, with a characteristic probability distribution determined by the quantum state of the system (the 'Born rule').

The theoretical description of such experiments relies on an appropriate coupling of the quantum system to an ancillary system, followed by a special application of the collapse and Born rule to the ancilla. In this way, the most general (physically sensible) families of transformations on the state of the quantum system can be implemented. Mathematically, this relies on the Naimark dilation theorem. (See for instance Busch, Grabowsky and Lahti (1995).)

This theoretical description was developed (for special cases) already by von Neumann (1932), and it contains a notorious regress: the application of the phenomenological rules is shifted from the quantum system to be measured to the ancilla; from the ancilla (if the special application of the rules is again modelled theoretically) it is shifted to a 'second-order' ancilla; from thence to a 'third-order' one, and so forth.

It was von Neumann himself who asked (at least for the sake of argument) whether this regress could be stopped if, at some stage, the apparent probabilistic aspect of the evolution could be explained as a statistical effect of coupling our system(s) of interest to some system of which only a coarse-grained description is available (a macrostate). Knowing the exact microstate of this system (and the exact form of the unitary evolution of the composite) would allow us to predict exactly the final state of the composite (say, a product consisting of the 'collapsed' state of the system and any microscopic ancilla(s), and a microstate of the apparatus belonging to a macrostate associated with a definite outcome). Thus, the theoretical description of the measurement process would be continuous and deterministic, but due to our ignorance of the exact microstate of the 'apparatus', the phenomenological one would be discontinuous and indeterministic, as given by collapse and Born rule. Von Neumann's argument to the effect that this redescription of the measurement process is impossible, and its various later generalisations, have become known as 'insolubility proofs' for the measurement problem of quantum mechanics. The intuitive core of von Neumann's argument is that if the statistical element of the measurement comes from the description of the apparatus, then it will not depend on the state of the system in the way required by the Born rule.

The later versions of von Neumann's theorem all focus on showing the impossibility in general of obtaining definite measurement results by assuming an initially mixed state for the apparatus. The simplest and most general of these is the one given by [START_REF] Bassi | A General Argument against the Universal Validity of the Superposition Principle[END_REF]. (See Brown (1986) and [START_REF] Bacciagaluppi | Insolubility Theorems and EPR Argument[END_REF] for further references and for discussion.)

In this paper, I shall provide an equally simple proof of insolubility, but following a different strategy, namely pointing out that no unitary interaction with an ancilla can deterministically steer the measured system to different 'collapsed' states, on pains of violating the quantum mechanical no-signalling theorem [START_REF] Ghirardi | A General Argument against Superluminal Transmission through the Quantum Mechanical Measurement Process[END_REF]. The result, which improves on my [START_REF] Bacciagaluppi | Insolubility Theorems and EPR Argument[END_REF], is an obvious consequence of the no-signalling theorem, but has never to my knowledge been brought out in this form.

In section 1 of the paper, I shall briefly rehearse in somewhat more detail the standard description of measurement based on the dilation theorem, using Stern-Gerlach measurements as an example. In section 2, I shall describe von Neumann's original insolubility argument, and in section 3 spell out my new argument. Brief remarks on related questions conclude the paper (section 4).

Stern-Gerlach measurements

Let us first take the case of an ideal measurement of spin. A system with both spin and position degrees of freedom is described using the tensor product of the Hilbert spaces used to describe a 'pure' spin-1/2 system and a spinless particle, just as if one were composing two separate systems.

Take an electron that we assume to be initially in a state |ψ ⊗ ϕ .

(1)

Now suppose we want to perform a measurement of spin in some given direction, and that with respect to this spin basis, |ψ = α|+ + β|-, so that (1) equals α|+ ⊗ ϕ + β|-⊗ ϕ .

(2)

If we pass the electron through an ideal Stern-Gerlach magnet, we obtain

α|+ ⊗ ϕ + + β|-⊗ ϕ - (3)
(where ϕ + and ϕ -are suitably deflected versions of ϕ). We see that the spin degree of freedom of the electron is entangled with its position degrees of freedom.

We now detect the electron on a screen, i.e. perform a position measurement on the spatial degrees of freedom of the electron. It is a discretised measurement of position, because we only need to distinguish whether the electron hits the half of the screen associated with the up or down component of the spin.1 Assuming that ϕ + and ϕ -do not overlap, the standard collapse postulate and Born rule, applied to the detection of the electron on the screen, will yield

|± ⊗ ϕ ± (4)
with probabilities |α| 2 or |β| 2 , respectively, and we have derived the collapse postulate and Born rule for the spin measurement from the collapse postulate and Born rule for the discretised measurement of position.

In reality, the theoretical description of the experiment will be more complex. For one thing, while we can expect the bulk, call it ϕ ++ , of ϕ + to be concentrated on one half of the screen (and similarly the bulk of ϕ -), it will have tails spreading out to the 'wrong' half of the screen, say

ϕ + = ϕ ++ + ϕ +- ( and
ϕ -= ϕ --+ ϕ -+ ). ( 5
)
Assuming for simplicity

|ϕ +-| 2 = |ϕ -+ | 2 =: ε and |ϕ ++ | 2 = |ϕ --| 2 = 1 -ε , (6)
then applying the collapse postulate and Born rule to detecting the electron on the screen yields

α|+ ⊗ ϕ +± + β|-⊗ ϕ -± (7)
(to be suitably renormalised), with probabilities

|α| 2 (1 -ε) + |β| 2 ε or |α| 2 ε + |β| 2 (1 -ε) , (8) respectively.
This yields a much more general transformation of the spin state of the electron (which is the degree of freedom we are measuring). Indeed, the spin of the electron is still entangled with its spatial degrees of freedom, and the spin part of the electron is collapsed to an improper mixture: either

|α| 2 (1 -ε)|+ +| + |β| 2 ε|--| (9) or |α| 2 ε|+ +| + |β| 2 (1 -ε)|--| (10)
(suitably renormalised).

One realises that even more general transformations on the spin state of the electron can be induced by a detection on the screen, if one considers that the Stern-Gerlach magnetic field itself is not 'ideal' (in order to satisfy the Maxwell equations, it cannot be perfectly homogeneous in the directions perpendicular to that of measurement). Or, indeed, if one considers that one could have chosen, at least in principle, any other unitary coupling between the spin and position degrees of freedom of the electron before proceeding to the detection on the screen. This illustrates the general phenomenology of measurements in quantum mechanics, as well as their general theoretical description: any physically sensible transformation on the state of the system (a 'completely positive' map) can always be obtained by suitable interaction with some ancilla (whether microscopic or macroscopic), followed by an application of the standard collapse postulate and Born rule on the ancilla. (As in the example, what is tested on the ancilla need not be one-dimensional projections.)2

Von Neumann's insolubility argument

In 1932, von Neumann published his famous discussions of the measurement process, in chapter VI of his Mathematical Foundations of Quantum Mechanics (von Neumann 1932). 3 The first two sections set up the problem and discuss composite systems, then von Neumann opens section 3 with announcing an argument 'to exclude an often proposed possibility for explaining the statistical character of process 1' (p. 437). Von Neumann's 'process 1' (defined on p. 351) is the collapse upon measurement of the quantum state of a system.

The explanation von Neumann is targeting appears to be the following. Assume that the measuring apparatus is initially in some appropriate mixed state ρ. For the sake of a vivid example, we could imagine that instead of coupling the spin of the electron to the spatial degrees of freedom of the electron, we couple it directly to a macroscopic 'pointer', and that initially we are not able to prepare a pure wavefunction ϕ for the pointer, but only a proper mixture

ρ = Λ w λ ρ λ dλ , (11)
where each ρ λ corresponds itself to a wavefunction ϕ λ localised within the spread of the original ϕ. For any initial (pure) state |ψ ψ| of the system, the final state evolving unitarily from the initial |ψ ψ| ⊗ ρ will also be mixed:

U(|ψ ψ| ⊗ ρ)U -1 . ( 12
)
The question is: could our ignorance of the initial state of the apparatus explain the final mixture of apparently collapsed states?

This idea (before he proceeds to refute it) is formulated by von Neumann as follows (pp. 437-438):

Let I be the observed system, II the observer. [...] The statistical character comes from the fact that while indeed I was in a homogeneous state 4 before the measurement, II instead was a mixture -and the character of mixture of II has 'infected' I + II during the course of the interaction, in particular has made a mixture of the projection in I. That is: the result of the measurement is indeterminate because the state of the observer before the measurement is not precisely known. Now, is it 'conceivable that such a mechanism works' (p. 438)? We would need to construct a Hilbert space H for the 'observer', together with a unitary evolution U for the total system, 5 an initial state ρ on H, and a finest ignorance-interpretable decomposition

ρ = Λ w λ ρ λ dλ (13)
of ρ, such that for any initial state

|ψ = α|+ + β|- (14)
of the system, there is a partition Λ = Λ + ∪ Λ -, which generally depends on |ψ , such that

Λ + w λ dλ = |α| 2 , Λ - w λ dλ = |β| 2 , (15)
and for all λ ∈ Λ ± ,

U(|ψ ψ| ⊗ ρ λ )U -1 = |± ±| ⊗ σ ± , (16)
where the σ ± represent distinct outcomes of the measurement.

Von Neumann's own argument to the contrary is that, assuming the relevant decomposition of the final state (12) has the form

w n |ψ n ψ n | ⊗ |ϕ n ϕ n | (17)
(where the |ψ n denote the eigenstates of the measured observable, and the |ϕ n are some orthonormal set of states in H), 6 the weights in this decomposition are uniquely determined by the weights in the orthogonal decomposition 5 Since the evolution of the total system is unitary, the total system needs to be closed, and we need to include in the 'observer' anything with which the system will interact directly or indirectly. That is, we need to include any ancillas, pieces of apparatus, and relevant parts of the environment (including human observers if applicable). For simplicity, we shall usually talk of the 'apparatus' or the 'screen', but we shall tacitly include in it everything else of relevance.

6 Presumably, the microscopic states corresponding to different readings need to be orthogonal, but will not form a basis of the (very high-dimensional) Hilbert space.

of the initial state ρ, and are thus independent of the state |ψ (in our example, this is inconsistent with ( 15)). This is true of any decomposition of this form, since the weights are unique up to ordering (pp. 438-439).7

3 Insolubility from no-signalling

We shall now show that the impossibility of obtaining ( 16) in general follows also (and very simply) from the quantum mechanical no-signalling theorem. The no-signalling theorem [START_REF] Ghirardi | A General Argument against Superluminal Transmission through the Quantum Mechanical Measurement Process[END_REF] states that performing a measurement on one of a pair of entangled particles (or any other manipulation, in particular unitary interaction with it) does not affect the probability distributions for results of measurements on the other. That is, while conditionalising on the outcomes of a measurement on one particle in general affects the probabilities for the outcomes of measurements on the other, conditionalising on performing the measurement (or any other manipulation) does not. For example, take two electrons in the singlet state. If we measure spin along the same direction on both particles, we have perfect anti-correlations between the outcomes, but averaging over the outcomes on one side yields back the usual 50-50 distribution on the other side. One easily sees that the same is true also for measurements along different spin directions on the two sides, etc.

(a) There exist a measurement scheme (H, U, ρ) and a finest ignorance-interpretable decomposition of ρ that explain the phenomenological collapse for any |ψ .

However, von Neumann's assumption that the ignorance-interpretable decomposition of the final state has the form (17) essentially constrains this decomposition uniquely, so that allowing the decomposition to depend on |ψ does not make the claim more general. That is, his proof rules out also the a priori weaker (b) There exists a measurement scheme (H, U, ρ) such that for any |ψ there exists a finest ignorance-interpretable decomposition of ρ that explains the phenomenological collapse.

Both [START_REF] Brown | The Insolubility Proof of the Quantum Measurement Problem[END_REF] and [START_REF] Bacciagaluppi | Insolubility Theorems and EPR Argument[END_REF] argue that, with the exception of [START_REF] Fine | Insolubility of the Quantum Measurement Problem[END_REF], [START_REF] Brown | The Insolubility Proof of the Quantum Measurement Problem[END_REF] and [START_REF] Bassi | A General Argument against the Universal Validity of the Superposition Principle[END_REF], the later generalisations of von Neumann's result rule out claims that are essentially equivalent to (b), and thus establish something stronger than insolubility in the original sense.

The name 'no-signalling' relates to the case in which the two entangled systems are thought of as spatially distant, but the theorem applies generally to any pair of degrees of freedom in an entangled state. To pick our example from section 1, performing a measurement of the position of an electron after it has left the magnetic field of a Stern-Gerlach apparatus (that is, after the dynamical coupling between the spin and the spatial degrees of freedom of the electron has ended) has no net effect on the probability distributions for any spin variables of the electron.

We shall stick to the Stern-Gerlach example, although the proof will be valid for the general case of measurements performed by letting first a microscopic ancilla interact with the system, and then performing some appropriate manipulation of the ancilla. It will be equally valid for the (less realistic) case in which we assume we can couple directly the observable to be measured with some macroscopic 'pointer', but the measurement is performed on one of a pair of entangled systems (e.g. on Alice's side of an EPR pair). Thus, the result will hold for a wide class of measurements, and will provide copious counterexamples to the claim that the statistical spread of results in a measurement can generally be explained by invoking ignorance of the exact initial state of some macroscopic system.

Assume we try and model, using just the unitary Schrödinger evolution, the interaction between the (microscopic) spatial degrees of freedom of the electron and the (macroscopic) screen, in such a way as to be able to derive the phenomenology of the standard collapse upon detection. In the ideal case, as we have seen, after the electron has passed through the Stern-Gerlach magnet, its state is given by (3), i.e.

|Ψ = α|+ ⊗ ϕ + + β|-⊗ ϕ -. (18)
We now apply the idea of ( 13)-( 16) to this case. We assume there is a decomposition of the state ρ of the screen,

ρ = Λ w λ ρ λ dλ , (19)
and a partition of the index set Λ,

Λ = Λ + ∪ Λ - (20)
(the statistics of results will actually not enter our argument, so that the further condition ( 15) is immaterial for our purposes), such that for all λ ∈ Λ ± ,

U(|Ψ Ψ| ⊗ ρ λ )U -1 = |± ±| ⊗ |ϕ ± ϕ ± | ⊗ σ ± , (21)
the σ ± representing macroscopic spots on the respective halves of the screen. 8

Now take any run of the measurement that has produced the result '+'. The component ρ λ of ρ that has led to this result is a state of the screen that will produce the result '+', if we let the electron impinge on it (through the appropriate unitary interaction). Similarly, there exists another state of a screen that will produce the result '-', if we let the elctron impinge on it via an appropriate unitary interaction. Of course, we cannot prepare these states of the screen in practice (we could prepare them in principle by reversing the time evolution). But we see that there exist states such that, by choosing to let the electron interact with these states by an appropriate unitary interaction, we can produce at will a spot on the upper or lower half of the screen. But now, given (21), we can also guarantee that the spin state of the electron 'collapses' to the corresponding eigenstate. (Or, if passage through the Stern-Gerlach magnet has produced the entangled state (7), we can guarantee that the spin state of the electron evolves to either ( 9) or ( 10), suitably renormalised.)

We see that in principle we can influence the statistics of further measurements on the spin degree of freedom of the electron. But this is impossible, because we have interacted unitarily only with the spatial degrees of freedom of the electron (coupling them to this or that state of the screen), and the no-signalling theorem prevents this from having any effect whatsoever on the spin state of the electron. Thus, the no-signalling theorem rules out the possibility that an imperfectly known initial state of the screen might explain the collapse of the spin 8 More generally, we can assume that the total system evolves to states that are indistinguishable from the states on the right-hand side of ( 21), even though they might contain some classical or quantum correlations. These states need not be the same on each run of the experiment, as long as they are recognisable as indicating the given result. We can also allow for slight variation (even co-variation) from one run of the experiment to another both in the initial states |Ψ and in the measurement scheme (H, U, ρ), as long as for the correct proportion of runs a condition of the form (21) holds. We can even include a further component Λ 0 in (20), giving rise to 'dud runs' of the experiment.

part of the state of the electron in any Stern-Gerlach measurement (whether ideal or not). Similarly, it rules out the possibility that an imperfectly known initial state of Alice's apparatus might explain the collapse of the state of Bob's particle in an EPR experiment. In sum, the collapse of the state of a quantum system cannot in general be explained through ignorance of the exact microstate of the apparatus, just as von Neumann established.

Concluding remarks

Although I believe the argument as stated is new, there is a family resemblance with a number of other arguments in the literature, in particular the argument showing that deterministic hidden variables theories must violate locality (in the form of parameter independence) in order to recover the quantum mechanical correlations. In the case of an EPR pair in the Bohm theory, one can see this very easily. Indeed, if Alice knew the initial position of her particle, by choosing appropriately either the sign of the gradient or the polarity of the field she could arrange her Stern-Gerlach magnet in such a way as to obtain spin up or spin down at will, and thus force the opposite result on Bob's side (see e.g. the nice description in [START_REF] Albert | Quantum Mechanics and Experience[END_REF], chapter 7). More generally, since deterministic hidden variables have to be contextual in general [START_REF] Kochen | The Problem of Hidden Variables in Quantum Mechanics[END_REF], in the case of perfect correlations the context dependence for one particle translates into nonlocality across the particle pair. Arguments of this kind are known as algebraic proofs of nonlocality (e.g. [START_REF] Stairs | Quantum Logic, Realism, and Value-definiteness[END_REF][START_REF] Heywood | Nonlocality and the Kochen-Specker Paradox[END_REF][START_REF] Bacciagaluppi | Critique of Putnam's Quantum Logic[END_REF]. Contrapositively, they yield proofs of indeterminism from the assumption of locality [START_REF] Conway | The Free Will Theorem[END_REF]. Note that in this paper we have assumed unitary dynamics throughout and no hidden variables, so locality follows from the no-signalling theorem and need not be assumed separately.

Similar arguments hold also for theories with super-quantum correlations as long as they satisfy the no-signalling condition, e.g. in the case of Popescu-Rorlich boxes, as pointed out by Jeffrey Bub in this same issue (Bub 2012a; see also [START_REF] Bub | Bananaworld: Quantum Mechanics for Primates[END_REF].

Finally, I wish to point out the resemblance with Einstein's own version of the EPR argument (the comparison is made in more detail in Bacciagaluppi (2013)9 ). As discussed by [START_REF] Fine | Einstein's Critique of Quantum Theory: The Roots and Significance of EPR[END_REF] and [START_REF] Howard | Einstein on Locality and Separability[END_REF][START_REF] Howard | Nicht sein kann was nicht sein darf", or the Prehistory of EPR, 1909-1935: Einstein's Early Worries about the Quantum Mechanics of Composite Systems[END_REF]), Einstein's own version of the argument very simply considered two alternative measurements on sub-system A of an entangled pair, leaving B in a state of one of two different types. However, The 'separation hypothesis' used by Einstein of course is stronger than the assumption of no-signalling,11 and we can illustrate why this needs to be the case by reference to the insolubility scenario. Einstein states explicitly that, if the quantum mechanical description is complete,

[t]he statistical character of the measurement outcomes is accounted for exclusively in terms of the measurement apparatus, or the process of measurement.

If this merely meant the initial state of the measurement apparatus, then we would have von Neumann's situation. 12 As we have seen, Einstein could then help himself to what Alice could do in principle, and only the locality part of the separation principle would be needed for the contradiction. But if the 'measurement process' is meant to be more general, e.g. a genuinely indeterministic collapse process, then Alice cannot control the outcomes even in principle, and the full separation principle is needed (but one also only needs to consider what manipulations Alice can perform in practice). In either case, a theory satisfying the suitable locality or separability condition will fail to reproduce the phenomenological collapse of the state of the distant system (and is perhaps also in this sense 'incomplete').

  [t]he real state of B now cannot depend on what measurement I perform on A (Einstein to Schrödinger, 19 June 1935; all translations from this letter are by E. Crull and myself). 10 Thus, quantum mechancal states are not the 'real states' of a system.

Note this might or might not coincide with the upper or lower half of the screen, as it depends on both the sign of the gradient and the polarity of the magnetic field.

For a comprehensive discussion in the language of POV measures, see e.g.Busch, Grabowsky and Lahti (1995).

Page references are to the 1955 translation (which I have mostly tacitly amended).

The 1955 translation has 'I was in a (unique) state', but the German 'einheitlich' is used by von Neumann explicitly as a synonym for 'rein', i.e. 'pure', in which case it is generally translated as 'homogeneous' (see e.g. the definition of a 'homogeneous or pure' expectation functional on p. 307). Clearly, von Neumann here means a pure state, represented by a projection |ψ ψ|.

Note that, arguably, the claim von Neumann is explicitly ruling out has the form

Please be aware of a confusing amalgam of text and quotation on pp. 95-96 (introduced after the proofs were corrected).

AHQP, mf. 92, section 2-107; this letter has been recently published (in German) in von Meyenn(2011), vol. 2, pp. 537-539.

[START_REF] Howard | Einstein on Locality and Separability[END_REF] analysis, it corresponds to the conjunction of 'separability', i.e. the existence of separate states for the two subsystems, and 'locality', i.e. no-signalling, a distinction which Einstein starts making explicitly by 1946 or 1947.

In the classical analogue that precedes the quantum mechanical discussion in the letter, Einstein talks about 'insufficiently known factors foreign to the described system'.

Acknowledgements

I wish to thank audiences at Aberdeen, Berlin, Cagliari and Oxford, as well as Arthur Fine and Max Schlosshauer, who heard or read and commented on previous versions of this and connected material. I am particularly indebted to Alex Blum, Martin Jähnert and especially Christoph Lehner for discussions of Einstein's argument and of how it might relate (or not) to von Neumann's. These discussions also helped me redirect my use of the no-signalling theorem to the general case of measurements with ancillas, whether or not there is spatial separation. Finally, I wish to thank Elise Crull, my collaborator on the Leverhulme Trust Project Grant 'The Einstein Paradox': The Debate on Nonlocality and Incompleteness in 1935 (project grant nr. F/00 152/AN), during the tenure of which this paper was written.