Thermal neutron cross sections of amino acids from average contributions of functional groups
G. Romanelli, D. Onorati, P. Ulpiani, S. Cancelli, E. Perelli-Cippo, J. I. Márquez Damián, S.C. Capelli, G. Croci, A. Muraro, M. Tardocchi, G. Gorini, C. Andreani, R. Senesi
TThermal neutron cross sections of amino acids from average contributionsof functional groups
Giovanni Romanelli, Dalila Onorati, a) Pierfrancesco Ulpiani, Stephanie Cancelli, Enrico Perelli-Cippo, JoséIgnacio Márquez Damián, Silvia C. Capelli, Gabriele Croci,
4, 6
Andrea Muraro, Marco Tardocchi, GiuseppeGorini, Carla Andreani,
2, 7 and Roberto Senesi
2, 8 ISIS Neutron and Muon Source, UKRI-STFC, Rutherford Appleton Laboratory, Harwell Campus, Didcot, Oxfordshire OX11 0QX,United Kingdom Università degli Studi di Roma “Tor Vergata”, Dipartimento di Fisica and NAST Centre, Via della Ricerca Scientifica 1,Roma 00133, Italy Università degli Studi di Roma “Tor Vergata”, Dipartimento di Scienze e Tecnologie Chimiche, Via della Ricerca Scientifica 1,Roma 00133, Italy Università di Milano-Bicocca, Piazza della Scienza 3, Milano, Italy European Spallation Source ERIC, P.O. Box 176, 22100 Lund, Sweden Istituto per la Scienza e Tecnologia dei Plasmi, CNR, via Cozzi 53, 20125 Milano,Italy CNR-ISM, Area della Ricerca di Roma Tor Vergata, Via del Fosso del Cavaliere 100, 00133 Roma,Italy CNR-IPCF, Sezione di Messina, Viale Ferdinando Stagno d’Alcontres 37, Messina, 98158,Italy (Dated: 12 February 2021)
The experimental thermal neutron cross sections of the twenty proteinogenic amino acids have been measured over theincident-neutron energy range spanning from 1 meV to 10 keV and data have been interpreted using the multi-phononexpansion based on first-principles calculations. The scattering cross section, dominated by the incoherent inelasticcontribution from the hydrogen atoms, can be rationalised in terms of the average contributions of different functionalgroups, thus neglecting their correlation. These results can be used for modelling the total neutron cross sections ofcomplex organic systems like proteins, muscles, or human tissues from a limited number of starting input functions.This simplification is of crucial importance for fine-tuning of transport simulations used in medical applications, includ-ing boron neutron capture therapy as well as secondary neutrons-emission induced during proton therapy. Moreover,the parametrized neutron cross sections allow a better treatment of neutron scattering experiments, providing detailedsample self-attenuation corrections for a variety of biological and soft-matter systems.
I. INTRODUCTION
The study of the interaction of neutrons with matter has stillto become centennial, yet it has impacted the modern societyin a variety of ways, from fission reactors to the creation ofisotopes for medical care; from the treatment of cancer to thenon-invasive characterization of cultural-heritage artworks; aswell as the scientific investigation of the structure and dynam-ics of condensed-matter systems . In the latter case, spe-cial attention should be paid to hydrogen ( H) for it mani-fests the largest bound scattering cross section (82.03 barn)amongst the elements of the periodic table . For this reason,neutron scattering off hydrogen is particularly relevant, for ex-ample, for molecular spectroscopy and catalysis , hydrogenstorage , and cultural heritage .Fermi was the first to model the interaction potential be-tween slow neutrons (energies lower than few eV) and hy-drogenous solids . He explicitly discussed how the total crosssections, σ ( E ) , for neutrons with incident energy E , wouldchange by approximately a factor of 4 when moving from coldneutrons to epithermal ones. On the one hand, epithermal neu-trons (eV – keV) have energies much larger than the typical a) Corresponding Author: [email protected] binding energy of hydrogen in a crystal or molecular system,therefore resulting in a Compton-like scattering from an ap-proximately free nucleus . In this case, the scattering isdefined as elastic in the neutron + nucleus system, thus requir-ing the conservation of kinetic energy and momentum of bothparticles, and the total scattering cross section corresponds tothe free-nucleus value, σ f . On the other hand, cold neutrons( µ eV – meV) impinging on a cold solid sample would bounceoff an almost unmovable target. Therefore, as such neutronsdo not change their kinetic energy following the interaction,the scattering is elastic in the system composed solely by theneutron, and the total scattering cross section corresponds tothe bound-nucleus value, σ b = σ f ( + m / M ) , where m is themass of the neutron, and M the mass of the nucleus. Thiscorresponds, in the case of hydrogen, to σ b (cid:39) σ f .In the intermediate (thermal) energy region, the scatteringis elastic in neither of the two cases aforementioned, for partof the neutron kinetic energy can be transferred to rotationalor internal vibrations of a molecule, or lattice and to phononmodes in a crystal, in a regime generally referred to as inelas-tic neutron scattering . For crystalline samples, the cross sec-tion also depends on the structure via the presence of Braggedges. Moreover, while the picture drawn by Fermi holds ex-actly for solids at low temperatures, the total scattering crosssection at energies lower than tens of meV can be higher than σ b in a solid at room or higher temperature, following the a r X i v : . [ phy s i c s . c h e m - ph ] F e b population of the Stokes and anti-Stokes transitions related tothe vibrational modes, according to the Maxwell-Boltzmannstatistics. In a liquid, furthermore, contributions from diffu-sion motions can increase additionally σ ( E ) at values of E below some meV.Tabulated Thermal Cross Sections (TCS) are available forjust a handful of systems . In order to match the levelof detail in modern Monte Carlo nuclear transport codes,the possibility to calculate TCS from simplified models ,from molecular dynamics , as well as from ab initio calcu-lations has been a topic of recent development. Despitethe important effort redirected into the theoretical calculationof TCS, the experimental data available for comparison andvalidation of the models is quite scarce. Moreover, new exper-imental investigations, aimed to update decades-old data, areclearly needed, as recently demonstrated for the case of para-hydrogen . In this context, a sustained experimental pro-gramme has been started at the VESUVIO spectrometer at the ISIS Neutron and Muon Source (UK) , to investigateTCS of alcohols , organic systems , water and othermaterials used to moderate neutrons . The unprecedent-edly broad energy range for incident neutrons available at theinstrument , spanning from a fraction of meV to tens of keV,allows a complete characterization of total neutron cross sec-tions, from cold to epithermal neutrons. Moreover, the broadenergy range provides an accurate and self-consistent way tonormalize the experimental spectra for those samples whosedensity is more difficult to determine , such as powders orsamples experiencing in situ adsorption or phase transitions.In this framework, the measurement and calculation of TCSof a vast set of materials are challenging tasks, owing to theseveral non-trivial dependencies on the molecular structure,dynamics, and thermodynamic temperature. When the TCScalculation is applied to human tissues or muscles for appli-cations such as Boron Neutron Capture Therapy (BNCT) as well as to study secondary neutrons-emission induced dur-ing proton therapy , the possibility to reconstruct the crosssection of large proteins becomes challenging. In fact, the 20basic amino acids can combine to form tens of thousands up to several billions of proteins , making the task of eithercalculate or measure the entire set of related TCS unrealistic.Here we provide an experimental determination of the totalTCS of the twenty amino acids, as a function of the neutronenergy from a fraction of meV to tens of keV, together withfirst-principles calculations based on the incoherent approx-imation, thus particularly suitable for hydrogen-containingmaterials. Given such unprecedentedly consistent set of ex-perimental data, we attempt a rationalization of the TCS ofamino acids and, by extension, of proteins, as a set of averagecontributions of independent functional groups, such as CH n ,NH n , and OH. The process of average over the amino acidsallows to consider the functional groups as independent, andto obtain the final results directly by adding the different con-tributions from different functional groups, neglecting theircorrelations. We will refer to this procedure as the AverageFunctional Group Approximation, AFGA. This task would al-low the replication of complex TCS, e.g., for biophysical andmedical applications, having just a handful of initial parame- ters. II. MATERIALS AND METHODSA. Neutron experiment
Neutron-transmission experiments were performed at theVESUVIO spectrometer at the ISIS Neutron and MuonSource. All the 20 amino acids were commercially availablein their L-form from Sigma Aldrich as anhydrous powders.Samples were loaded as received in circular flat containerswith Nb faces perpendicular to the direction of the neutronbeam, and Al spacers defining the sample volume. The lattercorresponded to a cylindrical shape with thickness of either 1or 2 mm for all samples, and a circular area facing the beamwith diameter 5 cm, thus covering the entire circular beamprofile . All samples were measured at the temperature of300 K within the instrument’s closed-circuit refrigerator. Foreach sample, data were collected for about 0.5–1.5 hours, cor-responding to an integrated proton current of 90–270 µ Ahwithin the ISIS synchrotron. The values of mass, thicknessand integrated proton current are reported in Table I, togetherwith the molecular stoichiometry. Time-of-flight spectra of in-cident neutrons not interacting with the sample were obtainedusing the standard GS20 Li-doped scintillator available at theinstrument, as well as the newly installed double thick GasElectron Multiplier (GEM) detector . The latter was posi-tioned between the sample position and the GS20 monitor, ata distance from the moderator of 12.60 m, as calibrated at thebeginning of the experiment using the VESUVIO incident foilchanger . The GS20 transmission monitor and the samplepositions correspond to 13.45 m and 11.00 m, respectively .It is important to stress that, for most of the samples inves-tigated in the present experiment, transmission spectra fromboth the GS20 monitor and the GEM detector were availableat the same time, and provided concurrent measurements ofthe neutron cross section of the same sample using two inde-pendent equipments. Given the distances of both detectors,the counts due to multiple scattering events within the sampleare completely negligible as compared to experimental errorsbars .Moreover, the GEM detector allowed more precise mea-surements over the extended energy range available at the in-strument, down to 0.6 meV, corresponding to the so-calledempty neutron pulse at the Target Station 1 at ISIS , as com-pared to 3 meV of the GS20 detector. This is a consequence ofthe lower sensitivity to γ -rays in the GEM compared to scin-tillators like the traditional GS20 monitor.Transmission spectra were obtained, as a function of theincident neutron energy, using the Beer-Lambert law, as T ( E ) = α S ( E ) C ( E ) = exp ( − n σ ( E ) d ) , (1)where S ( E ) and C ( E ) are the spectra corresponding to sampleinside the container and empty container, respectively. More-over, n is the number density of molecules inside the sample Amino Acid Formula CH CH CH Zw. + Other σ f [barn] %H M [g] d [mm] Q [ µ Ah] StructureLeucine C H NO – 312.08 85.3 1.63 1.00 270 Ref. Isoleucine C H NO – 312.08 85.3 3.89 2.00 180 Ref. Valine C H NO – 266.40 84.6 3.41 1.00 270 Ref. Methionine C H NO S 1 2 1 NH S 267.36 84.3 2.97 2.00 90 Ref. Lysine C H N O – 4 1 NH NH Threonine C H NO OH 224.46 82.1 5.02 2.00 90 Ref. Alanine C H NO – 175.03 81.9 4.35 2.00 1080 Ref. Cysteine C H NO S – 1 1 NH SH 176.00 81.5 5.30 2.00 180 Ref. Serine C H NO – 1 1 NH OH 178.78 80.2 6.02 2.00 90 Ref. Proline C H NO – 3 1 NH – 225.44 80.1 2.25 1.00 180 Ref. Glycine C H NO – 1 – NH – 129.35 79.2 4.54 2.00 270 Ref. Arginine C H N O – 3 1 NH (NH ) , NH 362.63 79.1 3.46 1.00 270 Ref. Glutamic Acid C H NO – 2 1 NH OH 232.93 79.1 4.16 2.00 270 Ref. Phenylalanine C H NO – 1 6 NH Glutamine C H N O – 2 1 NH NH Tyrosine C H NO – 1 5 NH OH 289.00 77.9 2.12 2.00 180 Ref. Aspartic Acid C H NO – 1 1 NH OH 187.23 76.6 5.05 2.00 270 Ref. Asparagine C H N O – 1 1 NH NH Tryptophan C H N O – 1 6 NH NH 325.23 75.6 3.05 2.00 270 Ref. Histidine C H N O – 1 3 NH NH 250.20 72.7 4.69 2.00 270 Ref. TABLE I. Chemical formula and decomposition of L-amino acids, together with the total value of the free scattering cross section per formulaunit, σ f , as well as the relative amount of hydrogen atoms per formula unit. The amino acids are generally found in their Zwitterion form, thecation is reported in the column labelled Zw. + , and the formula unit is completed by adding the anion, COO − . For each sample are reportedthe values of mass ( ± ± ab initio simulations is also reported in the last column. volume, σ ( E ) is their energy-dependent total cross section,and d the thickness of the container. Finally, α is a normal-ization factor taking into account the different durations of themeasurements with and without the sample. In the case of theGS20 monitor, α is an energy-dependent normalization pro-vided by the measurement of the incident neutron beam by theinstrument monitor at 8.57 m from the moderator, before thesample position. In the case of the GEM detector, both S ( E ) and C ( E ) were normalized to the number of proton pulsescounted by the detector electronics. However, as a slight fluc-tuation of the efficiency with time is known to affect the GEMdetector , of magnitude ca. 2% with no dependence upon theneutron energy, the resulting transmission spectra were scaledso as to overlap to the GS20 spectra for epithermal neutrons. B. Calculations of thermal cross sections
Within the incoherent approximation , the double dif-ferential scattering cross section can be expressed as a sumof single-particle contributions weighted by the sum of coher-ent and incoherent bound scattering cross sections for eachisotope j , σ b , j , namely d σ d µ dE (cid:48) = (cid:114) E (cid:48) E ∑ j N j σ b , j S j ( (cid:126) Q , ω ) , (2)where µ is the cosine of the scattering angle, E and E (cid:48) are theinitial and final neutron energies, N j represents the stoichiom- etry of the sample, and S j ( (cid:126) Q , ω ) is known as scattering law (orscattering kernel) as function of the energy, ¯ h ω = E − E (cid:48) andmomentum, (cid:126) Q , transfers. This approximation is particularlysuitable for hydrogen-rich materials, as in the present case,for the incoherent scattering contribution from hydrogen sumsup to ca. 99% of the total. Moreover, we adopt the powder-average approximation for non-oriented samples, thus we willconsider the scattering law as a function of the modulus of themomentum transfer, Q , but not of its direction. S j ( Q , ω ) is made explicit by defining the Fourier transformof the intermediate self-scattering function. Sjolander showedthat, in a harmonic cubic Bravais lattice and within the Gaus-sian approximation, the scattering law can be expressed as S j ( Q , ω ) = e − W j ∞ ∑ n = ( W j ) n n ! H n ( ω ) , (3)where H ( ω ) = δ ( ω ) , H ( ω ) = g j ( ω ) ¯ h ω γ j ( ) (cid:20) coth (cid:18) ¯ h ω k B T (cid:19) + (cid:21) (4)and H n > ( ω ) = (cid:90) H ( ω (cid:48) ) H n − ( ω − ω (cid:48) ) d ω (cid:48) . (5)This procedure describes the so-called Multi-Phonon Expan-sion (MPE), where the n -th term represents the contributionfrom the scattering process involving n phonons. Thus, thefirst term gives the elastic cross-section; the next term givesthe cross-section for all one-phonon processes in which, inturn, each phonon is excited; and so on. The function g j ( ω ) represents the unit-area normalized Vibrational Density ofStates (VDoS) of a given nucleus, such that g j ( ω ) d ω is thefraction of the normal modes whose frequencies lie in therange between ω and ω + d ω . Moreover, in Eq. 3, γ j ( ) = (cid:90) ∞ g j ( ω ) ¯ h ω coth (cid:18) ¯ h ω k B T (cid:19) d ω , (6)is also used to define the so-called Debye-Waller factor,2 W j , with the relation2 W j = (cid:104) ( (cid:126) Q · (cid:126) u j ) (cid:105) = Q (cid:104) u j (cid:105) = E R , j γ j ( ) , (7)where E R , j = ¯ h Q / M j , (cid:104) u j (cid:105) is the mean square displace-ment, and the second equality holding within the isotropic ap-proximation.In principle, the MPE can be calculated up to any value of n , yet its numerical evaluation becomes progressively moretime-consuming. In fact, as the incident energy increases, anincreasingly large number of phonons is likely to be involvedin the scattering process. However, in the epithermal regionthe neutron scattering cross section can be interpreted in theImpulse Approximation (IA) and the scattering law can beexpressed as S IA , j ( Q , ω ) = exp (cid:16) − ( ¯ h ω − E R , j ) E R , j k B T j (cid:17)(cid:112) π E R , j k B T j , (8)where E R , j is interpreted, within the IA, as the nucleus re-coil energy, and T j is an effective temperature proportional tothe kinetic energy of the scattering nucleus. In this work, wehave adopted the transition energy between the MPE and theIA equal to 3 eV. In particular, above this energy and onlyfor the evaluation of the total scattering cross section, thevalue of T j can be approximated with T , the same thermo-dynamic temperature used in the MPE. Otherwise, at lowervalues of E , substantially higher values of T j should be used in the case of hydrogen well above room temperature (see e.g., Ref. ) and for heavier masses up to room temperature (see e.g.,
Refs. )The total scattering cross-section σ ( E ) is obtained by com-bining Eqs. 3 and 8 with Eq. 2, integrating over the valuesof µ , in the range [ − , ] , and over E (cid:48) . The VDoS can beobtained from first-principles computer simulations, as dis-cussed in the next section. From a numerical point of view,it is important to note that the limit of energy integration isonly formally extended to infinity because the integrand dif-fers from zero only in a finite interval up to the cut-off energy¯ h ω C = n ¯ h ω m , with ¯ h ω m the highest vibrational frequency inthe VDoS. Furthermore, each term H n ( ω ) is unit-area nor-malized. Finally, the total cross section is obtained by addingthe neutron absorption cross section. By neglecting any nu-clear resonance, not expected for H, C, O, S and N in therange investigated in this experiment, the absorption contri-bution has a simple form σ a , j ( E ) = σ a , j (cid:112) E / E , where the values of σ a , j , corresponding to the absorption at E = . . Therefore, σ ( E ) = (cid:90) d µ (cid:90) dE (cid:48) d σ d µ dE (cid:48) + ∑ j N j σ a , j ( E ) . (9) C. Phonon calculations
The VDoS used in the MPE to estimate the TCS was ob-tained from ab initio simulations using the Quantum Espresso(QE) code, with a 6 × × , while the crystal structures for each amino acidswere taken from the Refs. reported in Table I.A Density Functional Perturbation Theory (DFPT) calcu-lation was run on the optimized geometry, the output corre-sponded to a collection of eigenvalues of the molecular normalmodes, ω v , and eigenvectors representing each atomic motionfor a given normal mode, (cid:126) e v , j . The index v spans between 1and 3 N − N is the number of atoms inthe unit cell and where we neglected the three lowest-energytranslational modes. The atom-projected VDoS for atom j isobtained as g j ( ω ) = ∑ v δ ( ω − ω v ) e v , j , (10)where e v , j is the square modulus of the eigenvector of j -nucleus corresponding to the v -vibration. In particular, thequantities (cid:126) e v , j = (cid:112) M j (cid:126) u v , j are obtained from the dynamicalmatrix and are associated to the eigenvalue frequency ω v . Fi-nally, (cid:126) u v , j is the related atomic displacement, expressed as thedifference of the time-dependent and mean nuclear positions.Additional details regarding the calculation of eigenvectorsand eigenfrequencies were reported in Ref. .One should note that the ab initio simulations provide sub-stantial additional information with respect to the experiment,as one can analyse the VDoS for each nucleus in the moleculeseparately. Such functions can either be averaged over nucleiof the same element, such as H, N, C, S and O in the presentcase, or they can be averaged for nuclei in the same functionalgroup and over different molecules, such as H in a CH methylgroup. III. RESULTS
The experimental neutron transmission spectra of all aminoacids were interpreted according to Eq. 1. In particular, thevalue of − ln ( T ( E > )) was normalised to the constantfree scattering cross section of the molecule, σ f = ∑ j N j σ f , j , (11)where σ f , j is the free scattering cross section for isotope j ,and N j the related stoichiometry. This is a fundamental toolavailable at the VESUVIO spectrometer, for it allows a self-consistent normalization of the data not based on the measuredsample density, dependent on the powder grain size and pack-ing fraction, and quite different from the tabulated one for thebulk crystal. The values of the bound cross sections for H, O,C, N, and S were taken from , and we considered for each ele-ment the values averaged over the naturally abundant isotopes.Figure 1 shows the experimental total cross sections of(from top to bottom) glutamic acid (A) and aspartic acid (B);cysteine (C) and glycine (D); and valine (E) and alanine (F).For each sample, the results obtained using the GS20 andGEM monitors agree, within error bars, over the neutron en-ergy range spanning from 3 meV to 10 eV. For the investigatedenergies higher than 10 eV, up to 10 keV, all cross sectionsare found to be constant, as expected. Importantly, the newGEM detector allows a more precise determination of σ ( E ) in two key regions. First, more accurate data are obtained atenergies between 0.6 meV and 3 meV, using the empty-pulseat ISIS Target Station 1 where the spectra collected by theGS20 can be affected by an important environmental gammabackground , especially for samples with high scatteringpower. Yet, owing to the lower sensitivity to γ -rays providedby the GEM detector, the data collected with this new addi-tion are of exquisite quality. The second region where thenew detector provides a relevant improvement to the VESU-VIO instrument corresponds to epithermal neutrons. Here, thehigher efficiency of the detector allows a higher count rate,thus smaller experimental error bars and less scattering of thedata points around the value of the free scattering cross sec-tion. This is of particular importance for experiments involv-ing powder samples, as in this case, for it improves the nor-malization procedure based on Eq. 11 and discussed in Ref. .The experimental data in Figure 1 were compared to thepredictions from the MPE and IA, as discussed earlier. Inparticular, two models were tested for the scattering contribu-tions: model 1 (m1) applies the MPE to all elements in themolecule; while model 2 (m2) combines the result from theMPE applied to hydrogen only and the values of the free scat-tering cross sections, σ f , i for the elements i other than hydro-gen. As one can appreciate from the comparison in Figure 1,both models compare extremely well to the experimental data,and they differ only slightly one from the other. In particular,the values from model 1 are found to be, for all amino acids,slightly larger than those from model 2, as expected, and alsoslightly larger than the experimental data. One should notethat the MPE applied to the elements heavier than hydrogenis more sensitive to the lower-energy part of g i ( ω ) , where theatom-projected VDoS is more intense. Following the 1 / ω de-pendence in Eq. 6, one can expect that a slight error in thecalculation of the intensities of g i ( ω → ) becomes particu-larly evident on the final result, and especially at higher val-ues of T . It is also important to note that the calculation of σ ( E ) using model 2, i.e., applying the MPE only to hydrogenand using σ f , i for the other elements, provides a significantlysimpler approach and substantially lower computational costs,yet obtaining a very accurate result. Experimental results forthe neutron cross section of amino acids were previously pre- σ ( E ) [ b a r n ] − − − E [eV] (A) Glutamic Acid, (B) Aspartic Acid
GEM (exp)Calculated (m2)Calculated (m1)GS20 (exp) (A)(B) σ ( E ) [ b a r n ] − − − E [eV] (C) Cysteine, (D) Glycine
GEM (exp)Calculated (m2)Calculated (m1)GS20 (exp) (C)(D) σ ( E ) [ b a r n ] − − − E [eV] (E) Valine, (F) Alanine
GEM (exp)Calculated (m2)Calculated (m1)GS20 (exp) (E)(F)
FIG. 1. The total cross section of glutamic acid (A) and asparticacid (B); cysteine (C) and glycine (D); and valine (E) and alanine(F). For each sample, the experimental data from the GEM (blackerror bars) and GS20 monitors (green circles) are compared to thecalculated spectra using model 1 (red continuous line) and model 2(blue dashed line). sented in Ref. at one value of the incident neutron energyof 50 meV. We note that those results generally underestimatethe results from the present investigation, possibly because ofthe uncertainties on the density and thickness of the powdersamples avoided in our normalization procedure .Finally, we note that all calculated cross sections were ob-tained by a newly created Python algorithm, based on the for-malism explained in Section 2, and prepared in such a waythat it can be readily included in the MantidPlot environ-ment for the reduction and analysis of neutron experiments. Inparticular, the application of the script developed here wouldbe suitable for the reduction and normalization algorithm pre-sented in Ref. . While the agreement between our calcula-tions and the experimental results is a powerful benchmark ofour algorithm, we performed additional comparisons applyingthe LEAPR module of NJOY2016 to our VDoS. The resultsfrom our algorithm satisfactorily overlap with those obtainedusing NJOY2016, with a maximum difference in the case ofglycine at 1 meV of about 0.7%, and, therefore are not shownhere. IV. DISCUSSIONA. Average Functional Group Approximation
A final, fundamental, comment regarding Figure 1 is re-lated to the different features and energy dependence of thecross section of each amino acid. As mentioned earlier, thisis the result of the different VDoSs that make, in princi-ple, the determination of the cross section of proteins a chal-lenging task. In particular, pioneering studies using inelas-tic neutron scattering already showed how the VDoS (thusthe double-differential cross section, Eq. 2) of isolated aminoacids could differ significantly from those of the related dipep-tides . However, when integrating Eq. 2 so as to obtain thetotal cross section, some sharp features and differences vanish.In this framework, we have interrogated our ab initio simula-tions so as to extract the contribution to the total cross sectionfrom hydrogen atoms in different functional groups within themolecule.Figures 2 and 3 show the total cross section of hydrogen indifferent average functional groups, namely CH n , NH n , OH,and SH. In particular, the molecular structure of each aminoacid simulated was interpreted according to the separation infunctional groups reported in Table I. The total cross sectionfrom hydrogen atoms in the same or different amino acids butbelonging to the same type of functional group were averaged,and for each value of the incident neutron energy, a standarddeviation associated to this average was defined and corre-sponds to the error bars in the figures. The fact that signalsfrom the functional groups are found to differ beyond the cal-culated error bars is an a posteriori proof that the averagingprocedure was meaningful. One can appreciate, for examplelooking at the top panel of Figure 2, how the average contribu-tion to the total cross section in a methylidyne ( ≡ CH), methy-lene ( = CH ), and methyl groups ( − CH ) are markedly differ-ent one from the other. In particular, within the CH grouping, FIG. 2. Total cross section (top) and atom-projected VDoS (bottom)of hydrogen in CH n functional groups, averaged over all amino acidspresenting such groups. The VDoSs spectra are shifted in the y-scalefor the sake of clarity. one can distinguish the contributions from aliphatic CH andaromatic CH functional groups. This has to do with the dif-ferent VDoSs in the four groups, as shown in Figure 2 (bot-tom). For example, the Stokes and anti-Stokes low-energyexcitation and de-excitations corresponding to the CH rotorin the methyl group are more easily populated at room tem-perature, and provide a higher value of the total cross sectionsat thermal-to-cold neutron energies. On the other hand, as allVDoSs are normalized to one, the larger cross section of hy-drogen in a CH group at neutron energies lower than ca. 7meV has, as a counterpart, a depletion in the region betweenca. 7 meV and ca. 90 meV. The opposite is true for the hy-drogen in a methylidyne group, while hydrogen in a methy-lene group has intermediate values between the two previouscases. It is important to note that the average of cross sectionsover hydrogen atoms participating in same functional group isnot identical, in principle, to the cross section obtained fromthe average of VDoSs of the same hydrogen atoms. Howeverwe have checked that the results obtained following these twoapproaches, in the case of CH, CH and CH , are the samewithin our numerical accuracy. For this reason, we report inthe Supplementary Material the average VDoS of the func-tional groups considered so as to allow the calculation of thetotal cross sections at temperatures other then 300 K. FIG. 3. Total cross section of hydrogen in NH n , OH, and SH func-tional groups, averaged over all amino acids presenting such groups.In the inserts the respectively atom-projected VDoS. The possibility to express the total cross section of aminoacids as average contributions of independent functionalgroups allows, in principle, the possibility to accurately ap-proximate the cross section of a given protein by the a priori knowledge of its composition and the set of 9 functions repro-duced in Figures 2 and 3 and reported in the SupplementaryMaterial from 1 meV to 1 eV . In order to test this approxi-mation, we have compared the experimental cross sections ofthe 13 remaining amino acids with the predictions based onthe AFGA. The results, presented in Figure 4, are also com-pared with the calculations using model 2. In general, the pre-dictions compare extremely favourably to both experimentaldata and ab initio calculations based on model 2. Some differ-ences arise for neutron energies lower than few meV, where we find, in the case of lysine, a maximum difference of ca.5% at 1 meV between the prediction based on the AFGA andthe calculation based on model 2. As we have proved correctthe AFGA for a large set of amino acids, one can expect thatthe same approximation will hold as well for larger proteins.
B. Application to other organic materials
Triphenylmethane (C H ) CH is a material of great poten-tial interest as neutron moderator. This molecular compoundis composed by three phenyl groups connected through a cen-tral carbon atom. In the top panel of Figure 5, it is shown thecomparison between the experimental hydrogen cross sectionof triphenylmethane, measured on VESUVIO from Ref. ,and the predicted spectrum using the average cross sectionfrom aromatic CH functional group defined in our AFGAmodel. In the figure it is also reported the calculation, fromRef. , from NJOY2016 Nuclear Data Processing system us-ing as input the VDoS obtained by DFT simulations. We notethat the agreement of our results compared to experimentaldata and simulation is excellent, providing a successful test ofAFGA also for molecular systems different from amino acids.The bottom panel of Figure 5 shows the compari-son between the neutron scattering cross section of poly-methylmethacrylate, or lucite (C O H ) n , from Ref. andthe prediction based on the AFGA. To construct the totalcross section based on the AFGA, we considered lucite ascomposed by non-interacting chains of CH C(CH )CO CH units. The related DFT calculations from Ref. were per-formed using CASTEP and the output files were processedwith OCLIMAX to calculate the scattering law, then withNJOY2016 to calculate the neutron scattering cross sectionusing a procedure tested in other articles, e.g., Ref. . In Fig-ure 5 (bottom), the experimental data measured by Sibona etal. are also reported, as well as the cross section tabulatedin the ENDF/BVIII.0 library. A very good agreement be-tween the experimental data and the reconstruction using ourAFGA model has been found over the entire energy range,with a slight departure from the experimental data at neutronenergies below few meV. In particular, the AFGA seems toprovide a much better agreement than the results based ondirect DFT simulations in the broad region below ca. 100meV. To explain this result, one should consider that lucite is,in reality, a non-crystalline vitreous substance and, therefore,its simulation using DFT-based phonon calculations can bequite challenging, unless very large unit cells are considered.On the other hand, the AFGA provides the scattering contri-bution of a given functional group averaged over a series ofslightly different environments and, therefore, is more able toreproduce the disordered nature of the material at the atomicscale.The successful application of the AFGA to materials notrelated to amino acids of proteins is of particular interest forpossible applications to a broader family of systems, includingglassy materials, and prove its general applicability. In fact,while it is not expected for the AFGA to provide a high levelof accuracy for small crystalline systems, the averaging of σ ( E ) [ b a r n ] − − − E [eV] (A) Isoleucine, (B) Glutamine, (C) Serine
PredictedCalculated (m2)GS20 (exp) (A)(B)(C) σ ( E ) [ b a r n ] − − − E [eV] (A) Arginine, (B) Phenylalanine, (C) Threonine
GEM (exp)PredictedCalculated (m2)GS20 (exp) (A)(B)(C) σ ( E ) [ b a r n ] − − − E [eV] (A) Lysine, (B) Methionine, (C) Proline
PredictedCalculated (m2)GS20 (exp) (A)(B)(C) σ ( E ) [ b a r n ] − − − E [eV] (A) Tryptophan, (B) Tyrosine, (C) Histidine, (D) Asparagine
PredictedCalculated (m2)GS20 (exp) (A)(B)(C)(D)
FIG. 4. Total neutron cross section for the 13 amino acids not reported in Figure 1. The results obtained using the Average Functional GroupApproximation (AFGA, red line) are compared to the experimental results obtained using the GS20 (green circles) and GEM detector (blackerror bars). The results obtained applying model 2 (m2) to the specific phonon calculation of each molecule are also reported as blue dottedline. contributions of individual functional groups in a set of “train-ing” systems can be representative of both large molecule,polymers and disordered biophysical systems.
C. Modelling using a sigmoidal function
The definition of an average total cross section for hydrogenin organic systems, over the same energy range investigatedhere, was recently provided in Ref. . In that case, the hy-drogen total cross section was averaged over several organicmolecules, including β -alanine, urea, and tartaric acid, so asto take into account the average contributions over differentfunctional groups. In order to provide a simple analyticalmodel, the result was fitted with a logistic 4-parameter sig- moidal function of the form σ ( E ) ∼ = s f + s b + cE d . (12)Considering the additional insight that we have gathered inthis study, provided by the results of our ab initio calculationsand the possibility to separate the signal from different func-tional groups, we have applied the same analytic function toeach of 9 spectra obtained within the AFGA. The parametersobtained for all types of functional groups are shown in theTable II. They allow a better treatment of neutron scatteringexperiments, providing detailed sample self-attenuation cor-rections for a variety of biological and soft-matter systems. σ ( E ) [ b a r n ] − − − E [eV] (CH C(CH )CO CH ) n OCLIMAX + NJOYPredictedSibona et al.ENDF/B-VIII.0
FIG. 5. Hydrogen neutron cross section in solid triphenyl-methane(top) from Ref. and the prediction based on the AFGA. Total neu-tron scattering cross section of poly-methylmethacrylate or lucite(bottom) from Ref. and the prediction based on the AFGA.s f s b c dCH(ali) 19.14 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± V. CONCLUSIONS
We have provided experimental results for the total neu-tron cross section of the twenty proteinogenic amino acids inthe neutron energy range between 1 meV and 10 keV. Thesewere successfully reproduced applying the formalism of themulti-phonon expansion to the calculated vibrational densitiesof states obtained from ab initio simulations. Moreover, fromthe results of such calculations, we have defined the averagecontribution to the total cross section from hydrogen atoms indifferent functional groups. The set of 9 functions, definedby the average hydrogen dynamics in different chemical en-vironments, was found to accurately reproduce the total crosssection of all amino acids, in the framework named AverageFunctional Group Approximation, i.e.,
AFGA. Moreover, wefound that the AFGA could be successfully applied to triph-enylmethane and poly-methylmethacrylate, or lucite, a glassymaterial for which phonon calculations are known to have lim-ited success.Our results represent a considerable simplification in thechallenging task of reproducing the total neutron cross sectionof dipeptides and proteins, with applications to biophysics andmedicine. For example, our results can be utilised to providenew models for the macroscopic cross section of human tis-sues and muscles, and for applications to the cancer treatmentthrough boron neutron capture therapy.
VI. SUPPLEMENTARY MATERIAL
See the supplementary material for the values of the averagehydrogen total cross section at 300 K and average VDoSs forfunctional groups in the AFGA model.
VII. DATA AVAILABILITY
Raw data related to this article were generatedat the ISIS Neutron and Muon Source (UK), DOI:10.5286/ISIS.E.RB2010019. Derived data supportingthe findings of this study are available from the correspondingauthor upon reasonable request.
ACKNOWLEDGMENTS
The authors gratefully acknowledge the financial support ofRegione Lazio (IR approved by Giunta Regionale n. G10795,7 August 2019 published by BURL n. 69 27 August 2019),ISIS@MACH (I), and ISIS Neutron and Muon Source (UK)of Science and Technology Facilities Council (STFC); the fi-nancial support of Consiglio Nazionale delle Ricerche withinCNR-STFC Agreement 2014-2020 (N 3420), concerning col-laboration in scientific research at the ISIS Neutron and MuonSource (UK) of Science and Technology Facilities Council(STFC) is gratefully acknowledged.0The following article has been submitted to Journal ofChemical Physics. After it is published, it will be found athttps://aip.scitation.org/journal/jcp.
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