An Elementary Proof of Dodgson's Condensation Method for Calculating Determinants

In 1866, Charles Ludwidge Dodgson published a paper concerning a method for evaluating determinants called the condensation method. His paper documented a new method to calculate determinants that was based on Jacobi's Theorem. The condensation method is presented and proven here, and is demonstrated by a series of examples. The condensation method can be applied to a number of situations, including calculating eigenvalues, solving a system of linear equations, and even determining the different energy levels of a molecular system. The method is much more efficient than cofactor expansions, particularly for large matrices; for a 5 x 5 matrix, the condensation method requires about half as many calculations. Zeros appearing in the interior of a matrix can cause problems, but a way around the issue can usually be found. Overall, Dodgson's condensation method is an interesting and simple way to find determinants. This paper presents an elementary proof of Dodgson's method.Comment: 7 pages, no figure

Rapid Determination of Activation Energies for Gas-Phase Protein Unfolding and Dissociation in a Q-IM-ToF Mass Spectrometer

Ion mobility-mass spectrometry has emerged as a powerful tool for interrogating a wide variety of chemical systems. Collision-induced unfolding (CIU), typically performed in time-of-flight instruments, has been utilized to obtain valuable qualitative insight into protein structure and illuminate subtle differences between related species. CIU experiments can be performed relatively quickly, but unfolding energy information obtained from them has not yet been interpreted quantitatively. While several methods can determine quantitative dissociation energetics for small molecules, clusters, and peptides, these methods have rarely been applied to proteins, and never to study unfolding. Here, we present a method to rapidly determine activation energies for protein unfolding and dissociation, built on a model for energy deposition during collisional activation. The method is validated by comparing activation energies for dissociation of three complexes with those obtained using Blackbody Infrared Radiative Dissociation (BIRD); values from the two methods are in agreement. Several protein monomers were unfolded using CIU, including multiple charge states of both cations and anions, and activation energies determined. ΔH‡ and ΔS‡ values are found to be correlated, leading to ΔG‡ values that lie within a narrow range (~70–80 kJ/mol) and vary more with charge state than with protein identity. ΔG‡ is anticorrelated with charge density, highlighting the key role of Coulombic repulsion in gas-phase unfolding. Measured ΔG‡ values are similar to those computed for proton transfer within small peptides, suggesting that proton transfer is the rate-limiting step in gas-phase unfolding and providing evidence of a link between the Mobile Proton model and CIU

Experimental and Theoretical Investigation of Overall Energy Deposition in Surface-Induced Unfolding of Protein Ions

Recent advances in native mass spectrometry have enabled its use to probe the structure of and interactions within biomolecular complexes. Surface-induced dissociation, in which inter- and intramolecular interactions are disrupted following an energetic ion-surface collision, is a method that can directly interrogate the topology of protein complexes. However, a quantitative relationship between the ion kinetic energy at the moment of surface collision and the internal energy deposited into the ion has not yet been established for proteins. The factors affecting energy deposition in surface-induced unfolding (SIU) of protein monomers were investigated and a calibration relating laboratory-frame kinetic energy to internal energy developed. Protein monomers were unfolded by SIU and by collision-induced unfolding (CIU). CIU and SIU cause proteins to undergo the same unfolding transitions at different values of laboratory-frame kinetic energy. There is a strong correlation between the SIU and CIU energies, demonstrating that SIU, like CIU, can largely be understood as a thermal process. The change in internal energy in CIU was modeled using a Monte Carlo approach and theory. Computed values of the overall efficiency were found to be approximately 25% and used to rescale the CIU energy axis and relate nominal SIU energies to internal energy. The energy deposition efficiency in SIU increases with mass and kinetic energy from a low of -20% to a high of -68%, indicating that the effective mass of the surface increases along with the mass of the ion. The effect of ion structure on energy deposition was probed using multiple stages of ion activation. Energy deposition in SIU strongly depends on structure, decreasing as the protein is elongated, due to decreased effective protein-surface collisional cross section and increased transfer to rotational modes

Evaluating Volatile Organic Compound Emissions from Cross-Laminated Timber Bonded with a Soy-Based Adhesive

Volatile organic compound (VOC) emissions from indoor sources are large determinants of the indoor air quality (IAQ) and occupant health. Cross-laminated timber (CLT) is a panelized engineered wood product often left exposed as an interior surface finish. As a certified structural building product, CLT is currently exempt from meeting VOC emission limits for composite wood products and confirming emissions through California Department of Public Health (CDPH) Standard Method testing. In this study, small chamber testing was conducted to evaluate VOC emissions from three laboratory-produced CLT samples: One bonded with a new soy-based cold-set adhesive; a second bonded with a commercially available polyurethane (PUR) adhesive; and the third assembled without adhesive using dowels. A fourth commercially-produced eight-month-old sample bonded with melamine formaldehyde (MF) adhesive was also tested. All four samples were produced with Douglas-fir. The test results for the three laboratory-produced samples demonstrated VOC emissions compliance with the reference standard. The commercially-produced and aged CLT sample bonded with MF adhesive did not meet the acceptance criterion for formaldehyde of ≤9.0 µg/m3. The estimated indoor air concentration of formaldehyde in an office with the MF sample was 54.4 µg/m3; the results for the soy, PUR, and dowel samples were all at or below 2.5 µg/m3

Increasing Collisional Activation of Protein Complexes Using Smaller Aperture Source Sampling Cones on a Synapt Q-IM-TOF Instrument with a Stepwave Source

Quadrupole-ion mobility-time-of-flight (Q-IM-TOF) mass spectrometers have revolutionized investigation of native biomolecular complexes. High pressures in the sources of these instruments aid transmission of protein complexes through damping of kinetic energy by collisional cooling. Since adducts are removed through collisional heating (declustering), excessive collisional cooling can prevent removal of non-specific adducts from protein ions, leading to inaccurate mass measurements, broad mass spectral peaks, and obfuscation of ligand binding. We show that reducing the source pressure using smaller aperture source sampling cones (SC) in a Waters Synapt G2-Si instrument increases protein ion heating by decreasing collisional cooling, providing a simple way to enhance removal of adducted salts from soluble proteins (GroEL 14-mer) and detergents from a transmembrane protein complex (heptameric Staphylococcus aureus α-hemolysin, αHL). These experiments are supported by ion heating and cooling simulations which demonstrate reduced collisional cooling at lower source pressures. Using these easily-swapped sample cones of different apertures is a facile approach to reproducibly extend the range of activation in Synapt-type instruments

A Simple Bioluminescent Method for Measuring D-Amino Acid Oxidase Activity

D-Amino acid oxidase (DAO) plays important roles in regulating D-amino acid neurotransmitters and was recently identified as a key enzyme integral to hydrogen sulfide production from D-Cys. We report here the development of a simple biocompatible, bioluminescent method for measuring DAO activity based on the highly selective condensation of D-Cys with 6-hydroxy-2-cyanobenzothiazole (CBT-OH) to form D-luciferin

Collidoscope: An Improved Tool for Computing Collisional Cross Sections with the Trajectory Method

Ion Mobility-Mass Spectrometry (IM-MS) can be a powerful tool for determining structural information about ions in the gas phase, from small covalent analytes to large, unfolded, and/or denatured proteins and complexes. For large biomolecular ions, which may have a wide variety of possible gas-phase conformations and multiple charge sites, quantitative, physically explicit modeling of collisional cross sections (CCSs) for comparison to IMS data can be challenging and time-consuming. We present a “trajectory method” (TM) based CCS calculator, named “Collidoscope”, which utilizes parallel processing and optimized trajectory sampling, and implements both He and N2 as collision gas options. Also included is a charge-placement algorithm for determining probable charge site configurations for protonated protein ions given an input geometry in pdb file format. Results from Collidoscope are compared to those from the current state-of-the-art CCS simulation suite, IMoS. Collidoscope CCSs are typically within 4% of IMoS values for ions with masses from ~18 Da to ~800 kDa. Collidoscope CCSs using x-ray crystal geometries are typically within a few percent of IM-MS experimental values for ions with mass up to ~3.5 kDa (melittin), and discrepancies for larger ions up to ~800 kDa (GroEL) are attributed in large part to changes in ion structure during and after the electrospray process. Due to its physically explicit modeling of scattering, computational efficiency, and accuracy, Collidoscope can be a valuable tool for IM-MS research, especially for large biomolecular ions

Multiple Evolutionary Origins of Ubiquitous Cu2+ and Zn2+ Binding in the S100 Protein Family

The S100 proteins are a large family of signaling proteins that play critical roles in biology and disease. Many S100 proteins bind Zn2+, Cu2+, and/or Mn2+ as part of their biological functions; however, the evolutionary origins of binding remain obscure. One key question is whether divalent transition metal binding is ancestral, or instead arose independently on multiple lineages. To tackle this question, we combined phylogenetics with biophysical characterization of modern S100 proteins. We demonstrate an earlier origin for established S100 subfamilies than previously believed, and reveal that transition metal binding is widely distributed across the tree. Using isothermal titration calorimetry, we found that Cu2+ and Zn2+ binding are common features of the family: the full breadth of human S100 paralogs—as well as two early-branching S100 proteins found in the tunicate Oikopleura dioica—bind these metals with μM affinity and stoichiometries ranging from 1:1 to 3:1 (metal:protein). While binding is consistent across the tree, structural responses to binding are quite variable. Further, mutational analysis and structural modeling revealed that transition metal binding occurs at different sites in different S100 proteins. This is consistent with multiple origins of transition metal binding over the evolution of this protein family. Our work reveals an evolutionary pattern in which the overall phenotype of binding is a constant feature of S100 proteins, even while the site and mechanism of binding is evolutionarily labile

Lipid Head Group Adduction to Soluble Proteins Follows Gas-Phase Basicity Predictions: Dissociation Barriers and Charge Abstraction

Native mass spectrometry analysis of membrane proteins has yielded many useful insights in recent years with respect to membrane protein-lipid interactions, including identifying specific interactions and even measuring binding affinities based on observed abundances of lipid-bound ions after collision-induced dissociation (CID). However, the behavior of non-covalent complexes subjected to extensive CID can in principle be affected by numerous factors related gas- subjected to extensive CID can in principle be affected by numerous factors related gas- subjected to extensive CID can in principle be affected by numerous factors related gas-subjected to extensive CID can in principle be affected by numerous factors related gas- subjected to extensive CID can in principle be affected by numerous factors related gas- subjected to extensive CID can in principle be affected by numerous factors related gas- subjected to extensive CID can in principle be affected by numerous factors related gas- phase chemistry, including gas-phase basicity (GB) and acidity, shared-proton bonds, and other factors. A recent report from our group showed that common lipids span a wide range of GB values. Notably, phosphatidylcholine (PC) and sphingomyelin lipids are more basic than arginine, suggesting they may strip charge upon dissociation in positive ion mode, while phosphoserine lipids are slightly less basic than arginine and may form especially strong shared-proton bonds. Here, we use CID to probe the strength of non-specific gas-phase interactions between lipid head groups and several soluble proteins, used to deliberately avoid possible physiological protein-lipid interactions. The strengths of the protein-head group interactions follow the trend predicted based solely on lipid and amino acid GBs: phosphoserine (PS) head group forms the strongest bonds with these proteins and out-competes the other head groups studied, while glycerophosphocholine (GPC) head groups form the weakest interactions and dissociate carrying away a positive charge. These results indicate that gas-phase thermochemistry can play an important role in determining which head groups remain bound to protein ions with native-like structures and charge states in positive ion mode upon extensive collisional activation

Eye lens β-crystallins are predicted by native ion mobility-mass spectrometry and computations to form compact higher-ordered heterooligomers

Eye lens crystallin proteins maintain the refractive properties of the lens but are not replaced after denucleation. Rolland et al. use native ion mobility-mass spectrometry, kinetics experiments, and computations to reveal that b-crystallins form heterodimers. These likely assemble into compact heterooligomers that enable the very high protein concentrations found in lens tissue