Recent advances in capillary electrophoresis‐mass spectrometry: Instrumentation,methodology and applications

Capillary electrophoresis (CE) offers fast and high‐resolution separation of charged analytes from small injection volumes. Coupled to mass spectrometry (MS), it represents a powerful analytical technique providing (exact) mass information and enables molecular characterization based on fragmentation. Although hyphenation of CE and MS is not straightforward, much emphasis has been placed on enabling efficient ionization and user‐friendly coupling. Though several interfaces are now commercially available, research on more efficient and robust interfacing with nano‐electrospray ionization (ESI), matrix‐assisted laser desorption/ionization (MALDI) and inductively coupled plasma mass spectrometry (ICP) continues with considerable results. At the same time, CE‐MS has been used in many fields, predominantly for the analysis of proteins, peptides and metabolites. This review belongs to a series of regularly published articles, summarizing 248 articles covering the time between June 2016 and May 2018. Latest developments on hyphenation of CE with MS as well as instrumental developments such as two‐dimensional separation systems with MS detection are mentioned. Furthermore, applications of various CE‐modes including capillary zone electrophoresis (CZE), nonaqueous capillary electrophoresis (NACE), capillary gel electrophoresis (CGE) and capillary isoelectric focusing (CIEF) coupled to MS in biological, pharmaceutical and environmental research are summarized.     

Revealing the potential of capillary electrophoresis/mass spectrometry: the tipping point

The hyphenation of capillary electrophoresis and mass spectrometry (CE/MS) remains a minor technique compared with liquid chromatography/mass spectrometry (LC/MS), which represents nowadays the standard instrumentation, regardless of its introduction thirty years ago. However, from a theoretical point of view, CE coupling should be quite favorable especially with electrospray ionization mass spectrometry (ESI‐MS). At the time, the sensitivity provided by CE/MS was often limited, due to hyphenation requirements, which at some point appeared to disqualify CE/MS from benefiting from the performance gain driving the evolution of MS instruments. However, this context has been significantly modified in a matter of a few years. The development of innovative CE/MS interfacing systems has enabled an important improvement regarding sensitivity and reinforced robustness in order to provide an instrumentation accessible to the largest scientific community. Because of the unique selectivity delivered by the electrophoretic separation, CE/MS has proved to be particularly relevant for the analysis of biological molecules. The conjunction of these aspects is motivating the interest in CE/MS analysis and shows that CE/MS is mature enough to enrich the toolbox of analytical techniques for the analysis of complex biological samples. Here we discuss the characteristics of the major types of high‐sensitivity CE/ESI‐MS instrumentation and emphasize the late evolution and future positioning of CE/MS analysis for the characterization of biological molecules like peptides and proteins, through some pertinent applications.     

Analysis of complex phthalic acid based polyesters by the combination of size exclusion chromatography and matrix-assisted laser desorption/ionization mass spectrometry

Matrix-assisted laser desorption/ionization time-of-flight mass spectrometry (MALDI-TOF MS) was used in conjunction with size exclusion chromatography (SEC) to investigate a model polyester system based on phthalic anhydride–1,2-propylene glycol. The polyesters were synthesized with a 30% molar excess of glycol, with kinetic samples being removed during different intervals of the polyesterification reaction. SEC was used to track the course of the reaction by determining the molecular weight and molecular weight distributions before subsequent off-line coupling with MALDI-TOF MS as a selective detection method to determine the chemical composition, identify the functionality type distributions as well as assist in assigning structural conformations. Mass spectrometry analysis proved to be a highly effective tool to facilitate the identification of the narrowly dispersed fractions obtained from the chromatographic separations as well as serve as a core method to investigate the heterogeneous nature of the bulk kinetic samples. Through the hyphenation of these sophisticated polymer characterization techniques, information on the molecular heterogeneity of the model polyesters, showing a complex variety of possible distributions, was obtained.     

Separation and Identification of Oligomeric-Hindered Amine Light Stabilizer Uvasorb HA 88 by High-performance Liquid Chromatography (HPLC) Coupled With Matrix-Assisted Laser Desorption/Ionization Time-of-Flight Mass Spectrometry (MALSI-TOF-MS)

Abstract

Uvasorb HA 88 is widely used as a light stabilizer to prevent plastic polymer degradation. Its high molecular weight and oligomeric characters provide challenge for quality control. In this study, high-performance liquid chromatography (HPLC) coupled with electrospray ionization time of flight mass spectrometry (ESI-TOF MS) was applied for separation and detection of Uvasorb HA 88. The synthesis scheme was deduced and confirmed by the characterization of Uvasorb HA 88 products from different batches. Direct matrix-assisted laser desorption/ionization time-of-flight mass spectrometry (MALDI-TOF-MS) was employed for the detailed characterization of chemical composition of Uvasorb HA 88. Furthermore, a reliable reversed-phase high-performance liquid chromatography (RP-HPLC) coupled with MALDI-TOF MS method was established and the weight-average and number-average molecular masses were calculated. The results revealed that six oligomers with repeat unit numbers from 1 to 4 were present in Uvasorb HA 88. A molecular weight of up to 6808?g mol^?1 was detected and two new series of oligomers were reported for the first time. Different series of oligomers and positional isomers were observed in Uvasorb HA 88. This work provides a suitable method to evaluate the technical grade of Uvasorb HA 88 as well as comprehensive characterization of complex oligomeric hindered amine light stabilizers.     

Modern mass spectrometry in the characterization and degradation of biodegradable polymers

In the last decades, the solid-waste management related to the extensively growing production of plastic materials, in concert with their durability, have stimulated increasing interest in biodegradable polymers. At present, a variety of biodegradable polymers has already been introduced onto the market and can now be competitive with non biodegradable thermoplastics in different fields (packaging, biomedical, textile, etc.). However, a significant economical effort is still directed in tailoring structural properties in order to further broaden the range of applications without impairing biodegradation. Improving the performance of biodegradable materials requires a good characterization of both physico-chemical and mechanical parameters. Polymer analysis can involve many different features including detailed characterization of chemical structures and compositions as well as average molecular mass determination. It is of outstanding importance in troubleshooting of a polymer manufacturing process and for quality control, especially in biomedical applications. This review describes recent trends in the structural characterization of biodegradable materials by modern mass spectrometry (MS). It provides an overview of the analytical tools used to evaluate their degradation. Several successful applications of MALDI-TOF MS (matrix assisted laser desorption ionization time of flight) and ESI MS (electrospray mass spectrometry) for the determination of the structural architecture of biodegradable macromolecules, including their topology, composition, chemical structure of the end groups have been reported. However, MS methodologies have been recently applied to evaluate the biodegradation of polymeric materials. ESI MS represents the most useful technique for characterizing water-soluble polymers possessing different end group structures, with the advantage of being easily interfaced with solution-based separation techniques such as high-performance liquid chromatography (HPLC).     

Weighing synthetic polymers of ultra‐high molar mass and polymeric nanomaterials: What can we learn from charge detection mass spectrometry?

Advances in soft ionization techniques for mass spectrometry (MS) of polymeric materials make it possible to determine the masses of intact molecular ions exceeding megadaltons. Interfacing MS with separation and fragmentation methods has additionally led to impressive advances in the ability to structurally characterize polymers. Even if the gap to the megadalton range has been bridged by MS for polymers standards, the MS‐based analysis for more complex polymeric materials is still challenging. Charge detection mass spectrometry (CDMS) is a single‐molecule method where the mass and the charge of each ion are directly determined from individual measurements. The entire molecular mass distribution of a polymer sample can be thus accurately measured. Described in this perspective paper is how molecular weight distribution as well as charge distribution can provide new insights into the structural and compositional studies of synthetic polymers and polymeric nanomaterials in the megadalton to gigadalton range of molecular weight. The recent multidimensional CDMS studies involving couplings with separation and dissociation techniques will be presented. And, finally, an outlook for the future avenues of the CDMS technique in the field of synthetic polymers of ultra‐high molar mass and polymeric nanomaterials will be provided.     

Interfacing multistage mass spectrometry with liquid chromatography or ion mobility separation for synthetic polymer analysis

Synthetic polymers are naturally mixtures of homologs, even in pure form. More complexity is introduced by the presence of different comonomers, end groups and/or macromolecular architectures. The analysis of such systems is substantially facilitated by interfacing mass spectrometry (MS), which disperses based on mass, with an additional level of separation involving either interactive liquid chromatography (LC) or ion mobility (IM) spectrometry, both of which are readily coupled online with electrospray ionization and MS detection. IM-MS separates in the gas phase, post-ionization and, therefore, is ideally suitable for labile and reactive polymers. Its usefulness is illustrated with the characterization of non-covalent siloxane-saccharide complexes, metallosupramolecular assemblies and an air- and moisture-sensitive inorganic polymer, poly(dichlorophosphazene). Conversely, LC-MS which separates in solution phase, before ionization, is most effective for the analysis of polymeric mixtures whose components differ in polarity. Interactive LC conditions can be optimized to disperse by the content of hydrophobic units, as is demonstrated for amphiphilic polyether copolymers and sugar-based nonionic surfactant blends. Both LC-MS and IM-MS can be extended into a third dimension by tandem mass spectrometry (MS(2)) studies on select oligomers, in order to obtain insight into individual end groups and isomeric architectures, comonomer sequences and degree of substitution, for example, by hydrophobic functionalities.     

Tandem mass spectrometry of synthetic polymers

The detailed characterization of macromolecules plays an important role for synthetic chemists to define and specify the structure and properties of the successfully synthesized polymers. The search for new characterization techniques for polymers is essential for the continuation of the development of improved synthesis methods. The application of tandem mass spectrometry for the detailed characterization of synthetic polymers using the soft ionization techniques matrix‐assisted laser desorption/ionization mass spectrometry (MALDI‐MS) and electrospray ionization mass spectrometry (ESI‐MS), which became the basic tools in proteomics, has greatly been increased in recent years and is summarized in this perspective. Examples of a variety of homopolymers, such as poly(methyl methacrylate), poly(ethylene glycol), as well as copolymers, e.g. copolyesters, are given. The advanced mass spectrometric techniques described in this review will presumably become one of the basic tools in polymer chemistry in the near future. Copyright © 2009 John Wiley & Sons, Ltd.     

On the use of different mass spectrometric techniques for characterization of sequence variability in genomic DNA

After completion of the human genome sequence the search for differences among individual genomes has become the centre of focus for geneticists. Two different types of polymorphism—single nucleotide polymorphisms (SNPs) and short tandem repeats (STRs)—are major sources of genetic diversity and are of widespread use in genetic analysis. A plethora of genotyping techniques have been developed, and mass spectrometry (MS) is among the most widely used analytical platforms. The most striking advantage of mass spectrometric genotyping assays over others is the use of the measured molecular mass information for allele calling. The molecular mass is less error-prone than other sequence-specific parameters, including migration times, retention times, or hybridization yields, as it represents an intrinsic property of a nucleic acid molecule that is directly related to its nucleotide composition. Mass spectrometric assays can roughly be divided into two major groups—matrix-assisted laser desorption/ionization (MALDI)-based and electrospray ionization (ESI)-based assays. An important subdivision of ESI-based genotyping methods are approaches that originate from the hyphenation of liquid chromatography (LC) to MS. The principles of these three classes of mass spectrometric genotyping techniques are summarized in this review. Possibilities and limitations are critically discussed to assist scientists, especially non-experts in MS, in choosing the appropriate mass spectrometric assay for genotyping a genetic marker of interest. Figure Comparison of the principle workflows applied for the characterization of genetic markers by MALDI–MS, ESI–MS, and LC–MS     

Real-time 2D separation by LC × differential ion mobility hyphenated to mass spectrometry

The liquid chromatography–mass spectrometry (LC-MS) analysis of complex samples such as biological fluid extracts is widespread when searching for new biomarkers as in metabolomics. The success of this hyphenation resides in the orthogonality of both separation techniques. However, there are frequent cases where compounds are co-eluting and the resolving power of mass spectrometry (MS) is not sufficient (e.g., isobaric compounds and interfering isotopic clusters). Different strategies are discussed to solve these cases and a mixture of eight compounds (i.e., bromazepam, chlorprothixene, clonapzepam, fendiline, flusilazol, oxfendazole, oxycodone, and pamaquine) with identical nominal mass (i.e., m/z 316) is taken to illustrate them. Among the different approaches, high-resolution mass spectrometry or liquid chromatography (i.e., UHPLC) can easily separate these compounds. Another technique, mostly used with low resolving power MS analyzers, is differential ion mobility spectrometry (DMS), where analytes are gas-phase separated according to their size-to-charge ratio. Detailed investigations of the addition of different polar modifiers (i.e., methanol, ethanol, and isopropanol) into the transport gas (nitrogen) to enhance the peak capacity of the technique were carried out. Finally, a complex urine sample fortified with 36 compounds of various chemical properties was analyzed by real-time 2D separation LC×DMS-MS(/MS). The addition of this orthogonal gas-phase separation technique in the LC-MS(/MS) hyphenation greatly improved data quality by resolving composite MS/MS spectra, which is mandatory in metabolomics when performing database generation and search.     

Controlling the positive ion electrospray ionization of poly(ethylene glycols) when using ultra-high-performance supercritical fluid chromatography-mass spectrometry

Poly(ethylene glycols) are complex polymers often added to pharmaceutical formulations to improve drug solubility and delivery. One of the main challenges when using chromatographic techniques coupled to mass spectrometry is the unselective ionization of poly(ethylene glycols) oligomers. Additionally, when the chain length is large enough, multiple charged species are formed, further complicating the mass spectra and processing. This study uses the advanced oligomer separation provided by supercritical fluid chromatography with a mass spectrometry approach that selectively ionizes poly(ethylene glycols) as ammoniated molecules to simplify data analysis and facilitate batch-to-batch comparisons. Several visual representations of the response of the ionization events based on the polymer molecular weight and the repeating unit were used to elucidate trends in ionization. Evaluation of the influence of the oligomer length and end-group on the electrospray ionization of the polymer allowed the development of a process to enable selective ionization for these complex polymers.     

High‐resolution GC/MS studies of a light crude oil fraction

The continuous development in analytical instrumentation has brought the newly developed Orbitrap‐based gas chromatography / mass spectrometry (GC/MS) instrument into the forefront for the analysis of complex mixtures such as crude oil. Traditional instrumentation usually requires a choice to be made between mass resolving power or an efficient chromatographic separation, which ideally enables the distinction of structural isomers that is not possible by mass spectrometry alone. Now, these features can be combined, thus enabling a deeper understanding of the constituents of volatile samples on a molecular level. Although electron ionization is the most popular ionization method employed in GC/MS analysis, the need for softer ionization methods has led to the utilization of atmospheric pressure ionization sources. The last arrival to this family is the atmospheric pressure photoionization (APPI), which was originally developed for liquid chromatography / mass spectrometry (LC/MS). With a newly developed commercial GC‐APPI interface, it is possible to extend the characterization of unknown compounds. Here, first results about the capabilities of the GC/MS instrument under high or low energy EI or APPI are reported on a volatile gas condensate. The use of different ionization energies helps matching the low abundant molecular ions to the structurally important fragment ions. A broad range of compounds from polar to medium polar were successfully detected and complementary information regarding the analyte was obtained.     

Inorganic trace analysis by mass spectrometry

Mass spectrometric methods for the trace analysis of inorganic materials with their ability to provide a very sensitive multielemental analysis have been established for the determination of trace and ultratrace elements in high-purity materials (metals, semiconductors and insulators), in different technical samples (e.g. alloys, pure chemicals, ceramics, thin films, ion-implanted semiconductors), in environmental samples (waters, soils, biological and medical materials) and geological samples. Whereas such techniques as spark source mass spectrometry (SSMS), laser ionization mass spectrometry (LIMS), laser ablation inductively coupled plasma mass spectrometry (LA-ICP-MS), glow discharge mass spectrometry (GDMS), secondary ion mass spectrometry (SIMS) and inductively coupled plasma mass spectrometry (ICP-MS) have multielemental capability, other methods such as thermal ionization mass spectrometry (TIMS), accelerator mass spectrometry (AMS) and resonance ionization mass spectrometry (RIMS) have been used for sensitive mono- or oligoelemental ultratrace analysis (and precise determination of isotopic ratios) in solid samples. The limits of detection for chemical elements using these mass spectrometric techniques are in the low ng g⁻¹ concentration range. The quantification of the analytical results of mass spectrometric methods is sometimes difficult due to a lack of matrix-fitted multielement standard reference materials (SRMs) for many solid samples. Therefore, owing to the simple quantification procedure of the aqueous solution, inductively coupled plasma mass spectrometry (ICP-MS) is being increasingly used for the characterization of solid samples after sample dissolution. ICP-MS is often combined with special sample introduction equipment (e.g. flow injection, hydride generation, high performance liquid chromatography (HPLC) or electrothermal vaporization) or an off-line matrix separation and enrichment of trace impurities (especially for characterization of high-purity materials and environmental samples) is used in order to improve the detection limits of trace elements. Furthermore, the determination of chemical elements in the trace and ultratrace concentration range is often difficult and can be disturbed through mass interferences of analyte ions by molecular ions at the same nominal mass. By applying double-focusing sector field mass spectrometry at the required mass resolution—by the mass spectrometric separation of molecular ions from the analyte ions—it is often possible to overcome these interference problems. Commercial instrumental equipment, the capability (detection limits, accuracy, precision) and the analytical application fields of mass spectrometric methods for the determination of trace and ultratrace elements and for surface analysis are discussed.     

A robust sheath-flow CE-MS interface for hyphenation with Orbitrap MS

The hyphenation of capillary electrophoresis with high-resolution mass spectrometry, such as Orbitrap MS, is of broad interest for the unambiguous and exceptionally sensitive identification of compounds. However, the coupling of these techniques requires a robust ionization interface that does not influence the stability of the separation voltage while coping with oxidation of the emitter tip at large ionization voltages. Herein, we present the design of a sheath-flow CE-ESI-MS interface which combines a robust and easy to operate set-up with high-resolution Orbitrap MS detection. The sheath liquid interface is equipped with a gold coated electrospray emitter which increases the stability and overall lifetime of the system. For the characterization of the interface, the spray stability and durability were investigated in dependence of the sheath-flow rate, electrospray voltage, and additional gold coating. The optimized conditions were applied to a separation of angiotensin II and neurotensin resulting in LODs of 2.4 and 3.5 ng/mL.     

Gas phase reactivity of isomeric arylglycosides towards amines. A chemical ionization mass spectrometry and tandem mass spectrometry study

Chemical ionization mass spectrometry (MS) and tandem mass spectrometry (MS/MS) experiments have been performed for the structural characterization and isomeric differentiation of two series of C- and O-linked arylglycosides with potential antioxidant activity. Different amines have been used for producing gas phase chemical ionization. Depending on their proton affinity and steric hindrance, adduct ions with different stability are formed. The most stable adducts are produced by ethylamine and they have been extensively structurally characterized by experimental and theoretical approaches. Energy resolved chemical ionization tandem mass spectrometric experiments have allowed unambiguous characterization and differentiation of both the anomers differing at the configuration of the glycosidic C(1) atom, and regio- and structural isomers at extremely low concentrations, typical of mass spectrometry. This study has shown that amine chemical ionization mass spectrometry and MS/MS are powerful and versatile tools for the structural characterization of arylglycosides.     

Review coupling of capillary electrochromatography to mass spectrometry

This review discusses the development of capillary electrochromatography (CEC) coupled to mass spectrometric (MS) detection over the last few years. Major topics addressed are instrumental setups employed and applications of this technology published in the recent literature. The instrumental section includes a discussion of the most commonly used interfaces for the hyphenation of CEC and MS as well as ionization techniques. Applications reviewed in this paper come from a variety of different fields such as the analysis of biomolecules like proteins, peptides, amino acids or carbohydrates, chiral separations or the analysis of pharmaceutical an their metabolites in a series of matrices.     

Separation techniques hyphenated to electrospray-tandem mass spectrometry in proteomics: capillary electrophoresis versus nanoliquid chromatography

Liquid chromatography (LC) nanoelectrospray-tandem mass spectrometry (MS/MS) is a key technology for the study of proteomics, with the main benefit to the characterization of sensitive peptides from complex mixtures. Capillary electrophoresis coupled to mass spectrometry (MS) has been taken into consideration sporadically due to the highly efficient separation and ability to handle low sample amount, yet classified as being less sensitive with respect to analyte concentration. The limitation in capillary zone electrophoresis (CZE) injection volumes can be overcome by on-line solid-phase extraction (SPE). Such an on-line SPE-CZE system was explored in combination with an ion trap (IT) mass spectrometer. Thus, it was possible to inject more than 100 microL sample solution on to the CZE capillary. Concentration limits of detection as low as 100 amol/microL were demonstrated for a peptide standard. This SPE-CZE-microelectrospray ionization (ESI)-MS/MS setup was compared directly to nanoLC/nanoESI using the same sample of a tryptic digest of bovine serum albumin (BSA) as a reference standard. Measurements were made on one IT mass spectrometer with identical acquisition parameters. Both chromatography systems enabled the separation and detection of low levels of peptides from a mixture of moderate complexity, with most peptides identified using both techniques; however, specific differences were obvious. The nanoLC-MS is about five times more sensitive than the CZE-MS, yet the difference was less pronounced than expected. The CZE-MS technique showed reduced loss of peptides, especially for larger peptides (missed cleavages) and is about four times faster than the nanoLC-MS approach.     

分子烙印亲和色谱与质谱联用实现中草药活性成分分离鉴定一体化

以骆驼蓬种籽中抗肿瘤活性化合物哈尔明及哈马灵的结构类似物哈尔满作为模 板,用非共价键法制备了对哈尔明及哈马灵具有强亲和性的分子烙印聚合物。此分 子烙印聚合物作为液相色谱固定相与大气压电离飞行时间质谱联用,直接分离鉴定 了草药骆驼蓬种籽甲醇粗提物中所含的哈尔明及哈灵两种抗肿瘤活性成分。实验结 果证明了通过分子烙印亲和色谱与质谱联用方法,快速有效地对中草药活性成分分 离鉴定是可能是。     

Rapid analysis of lubricants by atmospheric solid analysis probe–ion mobility mass spectrometry

Formulated lubricants are complex mixtures composed of base oil(s) and additives with various functions (detergents, corrosion inhibiter, antioxidant, viscosity modifiers, etc.). Because of the aliphatic nature of base oil and the chemical diversity of additives, the characterization of lubricant is currently a long and complex process. The comprehensive analysis of lubricant samples involves several techniques such as nuclear magnetic resonance, mass spectrometry, chromatography and infrared spectroscopy. The coupling of atmospheric solid analysis probe (ASAP) with ion mobility‐mass spectrometry (IM‐MS) has been shown to be an efficient tool for the characterization of complex mixture containing vaporizable polar to non‐polar compounds. This approach affords the coupling of a direct ionization technique that does not require sample preparation, with a bi‐dimensional separation method with high peak capacity. In this work, we show that ASAP‐IM‐MS is a suitable method for rapid and direct characterization of lubricant samples. Indeed, base oil and additives yielded, by ASAP, ions series which could be separated by IM‐MS. Molecular additives such as Zn‐dithiocarbamate, phosphite, thiophosphate and Alkyl diphenylamine were ionized as molecular ions [M]^+? or protonated molecules [M + H]⁺, depending of their polarity. In some cases, fragment ions were observed, confirming the additive identification. In addition, high molecular weight polymeric additives such as poly(alkyl methacrylate) (PAM) were pyrolized in the ASAP source leading to characteristic fragment ions. ASAP‐IM‐MS is shown to be a powerful tool for studying complex mixtures, allowing the first comprehensive analysis of lubricants in just a few minutes. Copyright © 2014 John Wiley & Sons, Ltd.     

Ion mobility spectrometry – Mass spectrometry coupling for synthetic polymers

This review covers applications of ion mobility spectrometry (IMS) hyphenated to mass spectrometry (MS) in the field of synthetic polymers. MS has become an essential technique in polymer science, but increasingly complex samples produced to provide desirable macroscopic properties of high‐performance materials often require separation of species prior to their mass analysis. Similar to liquid chromatography, the IMS dimension introduces shape selectivity but enables separation at a much faster rate (milliseconds vs minutes). As a post‐ionization technique, IMS can be hyphenated to MS to perform a double separation dimension of gas‐phase ions, first as a function on their mobility (determined by their charge state and collision cross section, CCS), then as a function of their m/z ratio. Implemented with a variety of ionization techniques, such coupling permits the spectral complexity to be reduced, to enhance the dynamic range of detection, or to achieve separation of isobaric ions prior to their activation in MS/MS experiments. Coupling IMS to MS also provides valuable information regarding the 3D structure of polymer ions in the gas phase and regarding how to address the question of how charges are distributed within the structure. Moreover, the ability of IMS to separate multiply charged species generated by electrospray ionization yields typical IMS‐MS 2D maps that permit the conformational dynamics of synthetic polymer chains to be described as a function of their length.