Radiat Environ Biophys DOI 10.1007/s00411-014-0518-9
ORIGINAL PAPER
Monte Carlo calculations of energy deposition distributions of electrons below 20 keV in protein Zhenyu Tan • Wei Liu
Received: 4 October 2013 / Accepted: 26 January 2014 Ó Springer-Verlag Berlin Heidelberg 2014
Abstract The distributions of energy depositions of electrons in semi-infinite bulk protein and the radial dose distributions of point-isotropic mono-energetic electron sources [i.e., the so-called dose point kernel (DPK)] in protein have been systematically calculated in the energy range below 20 keV, based on Monte Carlo methods. The ranges of electrons have been evaluated by extrapolating two calculated distributions, respectively, and the evaluated ranges of electrons are compared with the electron mean path length in protein which has been calculated by using electron inelastic cross sections described in this work in the continuous-slowing-down approximation. It has been found that for a given energy, the electron mean path length is smaller than the electron range evaluated from DPK, but it is large compared to the electron range obtained from the energy deposition distributions of electrons in semi-infinite bulk protein. The energy dependences of the extrapolated electron ranges based on the two investigated distributions are given, respectively, in a power-law form. In addition, the DPK in protein has also been compared with that in liquid water. An evident difference between the two DPKs is observed. The calculations presented in this work may be useful in studies of radiation effects on proteins. Keywords Energy deposition Electron range Protein Monte Carlo simulation Radiation effect
Z. Tan (&) W. Liu School of Electrical Engineering, Shandong University, Jinan 250061, Shandong, People’s Republic of China e-mail: [email protected]
Introduction Proteins are very important bioorganic compounds. Histones, a kind of small molecular alkaline protein, are crucial components of chromatin and play an important role in cell differentiation and replication. Especially, it has been shown that histone modifications can regulate the cellular differentiation (Heintzman et al. 2009), and specific combinations of histone modifications can also affect the accumulation and function of DNA damage repair factors (van Attikum and Gasser 2009). Therefore, the functions and physical structures of proteins have been a subject of long-lasting experimental and theoretical interest (Terwilliger et al. 1998; Lazaridis and Karplus 2000; Brown 2006; Heintzman et al. 2009). On the other hand, radiation studies on bio-molecules have also been done for a long time and are of main concern of proteins or DNA (Kempner 2011). In these studies, the energy deposition and penetration range of energetic electrons in protein are two important quantities that require evaluation. Biological inactivity and main chain scission are the principal radiation effects in proteins reported (Lidzey et al. 1995; Ronan et al. 1996; Kempner 2001, 2011; Berovic et al. 2002). Radiation inactivation, where a single ionization of the protein is sufficient to inactivate it, can be utilized to directly determine the mass associated with the function of a protein (Ronan et al. 1996). For this determination, the knowledge of the average energy deposited by each inelastic event is required. In order to identify the relevant physical, chemical and biological parameters responsible for a radiation effect at the molecular level, the energy depositions in related biological targets, such as DNA, nucleosome, and chromatin fiber, have been calculated, in an effort to seek correlations between local energy depositions and the observed biological effectiveness of
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different radiation qualities (Nikjoo et al. 1989, 1991, 2003, 2008). These calculations, except those for DNA, involve the energy deposition of radiation in proteins because nucleosome and thus chromatin are composed of proteins and DNA. So far, radiation effects on protein molecules have been studied using high-energy electrons above 20 keV (Rauth and Simpson 1964; Lowe and Kempner 1982; Harmon et al. 1985; Kempner 2001), and the interactions and ranges of electrons in protein have also been investigated. In studies of the damage induced in protein crystals by synchrotron radiation, the energy of the emitted secondary electrons is higher than 10 keV; these electrons interact further with protein crystal generating a large number of new secondary electrons with sub-keV energies. Due to this, the penetration depth of these secondary electrons in proteins needs to be estimated (Nave 1995; O’Neill et al. 2002). In the present work, both the distributions of energy depositions of electrons in semi-infinite bulk protein and the radial dose distributions of point-isotropic mono-energetic electron sources [i.e., the so-called dose point kernel (DPK)] in protein are calculated, in the energy range below 20 keV. The calculations are based on a Monte Carlo method described in a previous work (Tan et al. 2012). Here, the ranges of electrons are evaluated by extrapolating two distributions, respectively, and compared with the range calculated using inelastic cross sections for electrons in the continuous-slowing-down approximation (CSDA). The energy dependence of the extrapolated electron ranges is presented in the power-law form. In addition, the DPK in protein is compared with that in liquid water.
Materials and methods In the present simulations, the interactions of energetic electrons with an organic medium include mainly two types of basic interaction, i.e., elastic and inelastic scattering. Elastic scattering is related to the interaction with the nuclei of target atoms, while inelastic scattering results from the interaction with the electronic shells of target atoms. The model used in the present simulations was described in more detail previously (Tan et al. 2012). Here, only a brief outline of this model is given. Monte Carlo model: elastic scattering The Mott cross section based on the partial wave method has been widely applied in Monte Carlo simulations of low-energy electron transport in various media. In the present work, the numerical Mott differential cross sections from the best known NIST database (Jablonski et al. 2003) for the elastic scattering of electrons are used. The Mott
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differential cross section derived from the relativistic wave equation of Dirac can be expressed as dr ¼ jf ðhÞj2 þjgðhÞj2 ; dX
ð1Þ
with f ðhÞ ¼
1 1 X fðl þ 1Þ½expð2idl Þ 1 þ l½expð2idl1 Þ 1g 2ik l¼0
pl ðcos hÞ; gðhÞ ¼
1 1 X ½expð2idl1 Þ expð2idl Þp1l ðcos hÞ; 2ik l¼1
ð2Þ ð3Þ
where h is the elastic scattering angle, dl is the phase shift corresponding to the lth partial wave, hk is the electron momentum, and pl(cosh) and p1l ðcoshÞ are the ordinary and associated Legendre polynomials, respectively.For lowenergy electron transport in organic media, the elastic interactions of electrons with the atoms of the different types of elements composing the organic compound of interest must be simulated frequently. This costs a large amount of cpu time, and therefore, the mean elastic cross section of an electron interacting with an atom is used here to calculate the elastic scattering, to allow for a high simulation efficiency. The mean differential elastic cross section for each atom is given by dre X ni dri ¼ ; dX n dX i
ð4Þ
P where ni is the atom density, n ¼ i ni , and dri/dX is the Mott differential cross section corresponding to the ith element composing the organic compound investigated.The mean total elastic cross section for each atom is then calculated via re ¼
Zp
dre 2p sin hdh: dX
ð5Þ
0
It is clear that using the mean cross section for each atom, the electron elastic scattering in a compound can be simulated in the same manner as that for a single element, obviously simplifying the calculation process of elastic scattering. It should be pointed out, however, that the described method of calculating electron elastic scattering neglects any condensed-phase effects because the NIST data are for isolated atoms only (Jablonski et al. 2003), and thus, the method may still be in error below an energy of a few hundred eV. A possible way to implement condensedphase effects in the electron elastic scattering can be found in Ferna´ndez-Varea et al. (2005). Moreover, the NIST data
Radiat Environ Biophys
allow the electron energy ranging from 50 eV to 300 keV. Hence, in the present simulations, the calculation of electron elastic scattering is limited by a cut-off energy of 50 eV. Monte Carlo model: inelastic scattering The dielectric response theory has been widely used for the description of electron inelastic scattering in the Monte Carlo simulation of electron transport in different media ¨ ztu¨rk and Williamson 1993; Ding and Shimizu 1996; (O Ferna´ndez-Varea et al. 1996; Emfietzoglou 2003; Nikjoo et al. 2006; Dingfelder et al. 2008; Akkerman et al. 2009). Previously, a dielectric model for simulating the inelastic interaction of electrons with organic molecules has been presented (Tan et al. 2012). In this model, the electron inelastic differential cross section is given by d2 rin 1 ¼ dð hxÞdq p a0 NE Z1 0 x 1 hq2 0 0 Im½1=eðx Þd x x Cex dx0 ; xq 2m 0
ð6Þ where E is the kinetic energy of the incident electron, a0 is the Bohr radius, N is the molecule density (or atom density), hx and hq are the energy loss and the momentum transfer, respectively, from the incident electron into the medium described by dielectric response function e(q,x), Im[-1/e(x)] is referred to as the optical energy-loss function (OELF), and Cex is given by 2 Q Q þ Cex ¼ 1 ; ð7Þ Eþ hx0 hx Eþ hx 0 hx with Q = h2q2/2 m, where m is the mass of the electron. Recently, Emfietzoglou et al. (2012a) presented a review of optical-data models describing low-energy electron inelastic scattering in a medium based on the dielectric response theory. In their review, different algorithms extending the optical data to finite momentum transfer are discussed, and it is shown that the models of Penn (1987) and Ashley (1988) represent a type of extension algorithm and overcome the need for a parametric fit of extendedDrude function by replacing the summation over a finite number of extended-Drude functions by an integration over a model energy-loss function. As shown previously (Tan et al. 2004a), Eq. 6 represents a simplification of the Penn model (1987) which is better associated with Ashley’s work (1988). In addition, in Eq. 6, Cex is a factor introduced to take into account the exchange effect between the incident electron and the electrons in the medium in the Ochkur
approximation (Ochkur 1965; Ferna´ndez-Varea et al. 1993, 2005). As pointed out by Ferna´ndez-Varea et al. (1993), the exchange effect correction in optical-data models was first considered by Ashely and Anderson (1981), who used a heuristic approach, i.e., a Mott-like scheme. After that, the Mott-like scheme has been used by many authors in their optical-data models to calculate the inelastic scattering of electrons in the media (Ashely 1988, 1990; Dingfelder et al. 1998; Akkerman and Akkerman 1999; Tan et al. 2004a; Emfietzoglou and Nikjoo 2005). More recently, the exchange–correlation effects in optical-data models of the dielectric function have been investigated in detail for the first time for carbon target (Emfietzoglou et al., 2013), where the different optical-data models corrected by the exchange–correlation were analyzed. The approach presented by Emfietzoglou et al. (2013) is manageable, and thus, the use of the exchange–correlation correction in the optical-data models for calculating the inelastic scattering of electrons in other target materials is expected in the future. For E0 = hx0 and DE = hx, integrating Eq. 6 and following Ferna´ndez-Varea et al. (1996), the differential cross sections with respect to resonance energy E0 and energy loss DE, respectively, are obtained, i.e., drin df ðE0 Þ ¼ rone ðE0 Þ dE0 dE0
ð8Þ
and drone ¼ dDE
ZQþ
p e4 1 dðDE E0 QÞCex dQ; E DEQ
ð9Þ
Q
pffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi with Q ¼ ð E E DEÞ2 , where rone(E0 ) is the cross section for exciting a unit-strength oscillator with resonance energy E0 . Equations 8 and 9 will be used for determining resonance energy E0 and energy loss DE by random sampling in the simulation of electron inelastic scattering. The detailed procedure of simulating electron transport in an organic medium based on the model given above can be found in Tan et al. (2012), where the model was justified through a series of calculations and comparisons with published experimental and theoretical results. Here, as an example of using this model, Fig. 1 presents (a) the calculated electron backscattering coefficients for bulk PMMA (C5H8O2) and (b) the calculated energy distributions of transmitted electrons of PMMA film. In Fig. 1a, the experiments of Joy and Joy (1996) and the theoretical results from Murata et al. (1981) are also plotted for comparison. Figure 1a shows that in the energy range of E0 B 20 keV, the present calculations are in satisfactory agreement with the experiment and with the theoretical results. Figure 1b compares the energy distributions of transmitted electrons for PMMA film of thickness
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Radiat Environ Biophys 0.16
10
-13
protein σe σin
(a) PMMA 2
cross section (cm )
This work
0.12
Joy and Joy (1996)
ηB
Murata et al. (1981)
0.08
10
-14
10
-15
0.04
0
5
10
15
20
E0 (keV)
10
2
10
3
10
4
energy (eV)
16
Fig. 2 Total elastic and inelastic cross sections of electrons for an equivalent unit (C107H197N29O49S2)n of the considered protein as a function of energy
(b) PMMA, 20 keV Adesida et al. (1980)
dηT /dU
12
This work
8
4
0 0.5
1600 nm
0.6
0.7
0.8
0.9
1.0
U=E/E0 Fig. 1 Electron backscattering coefficient gB as a function of primary energy E0 for bulk PMMA (a); Energy distributions of transmitted electrons for PMMA film with thickness 1,600 nm at E0 = 20 keV (b)
1,600 nm at 20 keV from the present calculation with those from Adesida et al. (1980). The comparison indicates reasonable agreement between the two theoretical results.
Results and discussion Cross sections In the present work, the considered protein is the same as that used by Tan et al. (2006), with an equivalent unit (C107H197N29O49S2)n and a mass density of 1.3 g/cm3. For the considered protein, there is no available experimental optical data, and thus, its OELF was evaluated here with the use of the empirical approach (Tan et al. 2004a, b).
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Using the evaluated OELF, pre-calculated look-up tables were calculated for determining resonance energy E0 and energy loss DE. In addition, in the simulations, the number of simulated tracks was 2 9 105–59105 for each primary energy, and the cutoff energy of terminating a track was taken as 50 eV because of the limitation of the minimum energy of 50 eV for the elastic cross-sectional calculations, as stated above. The total elastic and inelastic cross sections describe the probability with which an incident electron interacting with the medium may be scattered elastically or inelastically. Figure 2 shows the resulting calculated total elastic and inelastic cross sections of electrons for an equivalent unit of the considered protein as a function of electron energy in the range below 20 keV. It can be seen from Fig. 2 that above about 200 eV the probability for inelastic scattering of an incident electron is larger than that for elastic scattering, while the reverse is true for energies below 200 eV. Energy deposition in semi-infinite bulk protein Using the Monte Carlo model given above, the depth distributions of the energy deposition for electrons with primary energies lower than 20 keV in semi-infinite bulk protein were calculated. The distributions are described by dE(z)/dz, where z is the depth into the protein. Figure 3 presents these distributions for various primary energies E0. It is clearly observed from Fig. 3 that a larger primary energy results in a larger penetration range, a deeper position of the peak, and a smaller value of the peak. The range of electrons which penetrate into a semiinfinite sample is a quantity of interest for practical
Radiat Environ Biophys 40
16
1 keV
protein
protein, E0=5 keV 12
dE/dZ (eV/nm)
dE/dZ (eV/nm)
30
3 keV
20
5 keV
10
8
ED
Rext
4
10 keV 20 keV
0 0 10
1
10
2
10
3
10
0
4
10
200
400
depth z (nm)
600
800
depth z (nm)
1:5 RED text ¼ 40:82E0
for 0:5keV E0 \1:5 keV;
ð10Þ
1:72 RED ext ¼ 37:01E0
for 1:5keV E0 20 keV;
ð11Þ
Using Eqs. 10 and 11, RED ext can be evaluated conveniently. For the determination of RED ext s, because the models for elastic and inelastic scattering used in the present
4
10
protein
3
10
ED
applications (Kotera et al. 1981; Bongeler et al. 1993; Akkerman et al. 2009). Specifically, the range of the electrons penetrating into protein is of importance for exploring the radiation damage to protein (Nave 1995; Kempner 2001; O’Neill et al. 2002). The range of electrons is characterized by the quantity R, for which there are different definitions. A detailed description of these different definitions can be found elsewhere (Akkerman et al. 2009; Emfietzoglou et al. 2012b). A usual definition of R is the so-called extrapolated range Rext (Akkerman et al. 2009). The extrapolated range can be determined by extrapolating the linear part of the depth distribution of the calculated energy deposition, dE(z)/dz, to the z-axis, as illustrated in Fig. 4, which presents dE(z)/dz as a function of depth z at E0 = 5 keV (also presented in Fig. 3). In this figure, the extrapolated range is denoted by RED ext . Employing this approach, obtained RED values as a funcext tion of primary energy E0 are shown in Fig. 5. The fact that the range of electron obey approximately the power law R ¼ AE0B , where A and B are two coefficients, has already been shown for other solids by experiments and theoretical calculations (Kotera et al. 1981; Bongeler et al. 1993; Kurniawan and Ong 2007; Akkerman et al. 2009). For the bulk protein investigated here, Fig. 5 demonstrates that the E0 dependence of RED ext (in unit nm) can be fitted by
Fig. 4 Depth distribution of energy deposition, dE(z)/dz, at E0 = 5 keV for bulk protein, and RED ext determined by extrapolating the linear part of dE(z)/dz to the z-axis
Rext (nm)
Fig. 3 Depth distributions of energy deposition in bulk protein for different primary energies E0
2
10
1
10
1
10
E0 (keV) Fig. 5 Extrapolated electron range RED ext energy E0 for bulk protein
as a function of primary
simulations may include errors at energies below 500 eV, the RED ext s for E0 \ 500 eV have not been considered. Note that, because the RED ext values did not follow a clear linear relationship in the log10 RED ext versus log10 E0 representation, for the whole energy range, the E0 dependence of RED ext was fitted separately for two energy regions. Dose point kernel calculation The radial dose distribution of point-isotropic mono-energetic electron sources, i.e., the DPK, in liquid water is
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Radiat Environ Biophys 10
0 4
10
protein
protein 0.5 keV
-1
10
DPK
Rext
3
Range (nm)
DPK
1 keV -2
10
3 keV 5 keV
10
2
10
-3
10
1
10
-4
10
0
1
10
10
2
10
3
10
1
radial distance (nm)
E0 (keV)
Fig. 6 Dose point kernel (DPK) profiles for different primary energies in protein
usually calculated for understanding the interaction between ionizing radiation and a biological medium. In the calculation, the starting point of each electron track is placed at the origin of coordinates, around which liquid water is divided into a series of spherical shells at Dr intervals from the origin, and then DPK is given by DPKðE0 ; rÞ ¼
DEðrÞ ; Ntr E0
ð12Þ
where Ntr is the number of simulated tracks and DE(r) is the deposited energy due to Ntr primary electrons in a concentric spherical shell with radius r and thickness Dr. Based on Eq. 12, DPK calculations for liquid water were performed using different Monte Carlo codes (Uehara et al. 1993; Emfietzoglou et al. 2003; Bousis et al. 2011). In this work, the DPK values in protein were calculated for energies below 20 keV. Figure 6 presents the calculated DPK values for selected values of E0, where Dr is set to 1 nm and Ntr is taken as 2 9 105 for 0.5 keV, 1 keV, and 3 keV, and as 5 9 105 for 5 keV. From Fig. 6, it is clear that the primary energy dependence of the DPK is similar to that of dE(z)/dz shown in Fig. 3. In practice, DPK represents the radial distribution of energy deposition of energetic electrons in the medium, and therefore, it is also called the fraction of absorbed energy (Uehara et al. 1993; Emfietzoglou et al. 2003). Due to this, and in a manner analogous to that indicated in Fig. 4 for dE(z)/dz, one can obtain an extrapolated range in the radial direction through the DPK for a given E0. Here, this range is denoted DPK by RDPK ext . Figure 7 gives Rext as a function of primary energy E0. Similarly as above, the primary energy dependence of RDPK ext can also be fitted by
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10
Fig. 7 Extrapolated electron range RDPK ext from DPK and the calculated penetration parameter \V[ in protein as a function of primary energy E0 1:51 RDPK ext ¼ 47:45E0 1:7 RDPK ext ¼ 42:85E0
for 0:5 keV E0 \1:5 keV; for 1:5 keV E0 20 keV;
ð13Þ ð14Þ
In this work, we have also simulated the penetration parameter \V[ of low-energy electrons in protein, where V is defined as the distance of the most distant electron transfer point relative to the starting point, and \V[ is the mean value of V. The calculated \V[ values are also displayed in Fig. 7 for comparison. It can be seen that\V[ is smaller than RDPK ext for a given E0. Another well-known definition of the electron range R is the electron mean path length. According to this definition, the electron range is usually indicated as RCSDA, and it is calculated in the CSDA by RCSDA ¼
ZE0
ðdE=dxÞ1 dE;
ð15Þ
Ecut
where -dE/dx is the collision stopping power (SP), and Ecut is the energy cutoff in the calculation of RCSDA . In this work, using -dE/dx based on the dielectric model described by Eq. 6 and taking Ecut as 50 eV, RCSDA was calculated in the energy range below 20 keV and DPK compared with RED ext and Rext , as shown in Fig. 8. Interestingly, from this comparison, it can be observed that DPK RCSDA is larger than RED ext , but smaller than Rext . The reason for RCSDA \RDPK ext may be attributed to the following: In the calculation of DPK, all the electrons starting from their point of origin contribute their energies in total to DPK. On the other hand, RCSDA is only a mean path length traveled
Radiat Environ Biophys 0.05
104
protein
protein liquid water: using empirical OELF using Emfietzoglou's OELF
0.04
103
DPK
Rext
ED
0.03
Rext
E0=1 keV
DPK
Range (nm)
RCSDA
0.02
102
0.01
101 0.00
100
0
101
by an energetic electron, and thus, it becomes small, ED compared to RDPK ext . In addition, RCSDA [ Rext is partly due to the fact that the contribution of the backscattered electrons to the linear part of dE(z)/dz is very small (Murata 1974). Water is usually used as a substitute of biological tissues in radiation biology (Nikjoo et al. 2006). Therefore, it is of significance to compare the DPK in protein with that in water. Employing the Monte Carlo model described here, DPK for liquid water was calculated. Figure 9 presents a comparison between DPKs in liquid water and in protein, respectively, at 1 keV, where the DPKs in liquid water were calculated by the use of the OELFs from the empirical evaluation (Tan et al. 2004a, b) and from the parameterized model of Emfietzoglou et al. (2005), respectively. It can be seen from Fig. 9 that the distribution range of the DPK in protein is obviously small, compared to that in liquid water. Therefore, when liquid water is used as a substitute for protein in calculations of energy depositions, this difference should be taken into account. In addition, the difference between the DPKs in liquid water calculated by using the OELFs from Tan et al. (2004a, b) and from Emfietzoglou et al. (2005), respectively, is very small. Actually, the distribution range of the DPK is mainly determined by the SP of the medium for electrons. A large SP results in a small distribution range. For the compounds, the non-relativistic Bethe formula describing the SP (in eV/ nm) can be written as (Bousis et al. 2011) dE qZ 1:166E ¼ 7850 ln ; ð16Þ dx ME I
40
60
80
radial distance (nm)
E0 (keV) Fig. 8 Comparison between different electron ranges in protein, i.e., DPK RCSDA, RED ext ; and Rext
20
Fig. 9 Comparison between DPKs in protein and liquid water, respectively. OELF optical energy-loss function; dotted line calculated with an OELF from Tan et al. (2004a, b); dashed line calculated with an OELF from Emfietzoglou et al. (2005)
where E is the electron energy (in eV), M is the molecular weight of the medium with Z electrons per molecule, q is the mean excitation the mass density (in g/cm3), and Iis energy (in eV). Using the evaluated OELF for protein and liquid water, are 71.4 eV and 72.5 eV, respectively the calculated Is (Tan et al. 2004a, 2006), being very close to each other. In addition, the ratio of (Z/M)protein using (C107H197N29O49S2)n to (Z/M)water is about 0.95. Due to this, the difference between the distribution ranges of the DPKs for protein and liquid water is mainly induced by their different mass densities (1.3 g/cm3 for protein and 1 g/cm3 for liquid water). In a previous work (Tan et al. 2006), the SPs of protein and DNA for electrons in the energy range below 10 keV were calculated, and it was shown that these are systematically smaller for DNA than for protein. Therefore, it can be expected that the distribution range of the DPK for electrons below 10 keV in DNA is larger than that in protein. However, it is still required to calculate in detail the distributions of the DPKs of low-energy electrons in DNA in future studies.
Conclusion In this work, systematic Monte Carlo calculations were performed in an effort to study energy deposition distributions of electrons in semi-infinite bulk protein and radial dose distributions of point-isotropic mono-energetic electron sources (DPK) in protein, for electron energies below
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Radiat Environ Biophys
20 keV. Evaluation of electron ranges was done by extrapolating the two calculated distributions, respectively. Electron mean path lengths in protein were also calculated by using electron inelastic cross sections in the CSDA and compared with the electron ranges evaluated from the two energy deposition distributions. Furthermore, using the present Monte Carlo method, the DPK in liquid water was calculated and compared with that in protein. The present work showed that in the considered energy range, the electron mean path length is about 2–8 % smaller than that evaluated from DPK, but it is about 6–7 % larger than that obtained from the energy deposition distributions of electrons in semi-infinite bulk protein. These results may provide a reference when estimating electron ranges in studies of radiation effects on proteins. In addition, energy dependences of the extrapolated electron ranges based on the two distributions were described in a power-law form for conveniently evaluating the electron range in proteins in future studies. Especially, the present calculations indicated that the distribution range of the DPK in protein is obviously small in contrast with that in liquid water, due mainly to their different mass densities. This difference should be taken into account when liquid water is used as a substitute for protein in calculations of energy depositions. Acknowledgments The authors are grateful to Drs. D. C. Joy and C. S. Joy of the University of Tennessee for providing their database of the compiled experimental data of electron backscattered coefficients. This work was supported by the Foundation of Ministry of Education of China under Grant No.20120131110012.
References Adesida H, Shimizu R, Everhart TE (1980) A study of electron penetration in solids using a direct Monte Carlo approach. J Appl Phys 52(11):5962–5969 Akkerman A, Akkerman E (1999) Characteristics of electron inelastic interactions in organic compounds and water over the energy range 20–10000 eV. J Appl Phys 86(10):5809–5916 Akkerman A, Murat M, Barak J (2009) Monte Carlo calculations of electron transport in silicon and related effects for energies of 0.02–200 keV. J Appl Phys 106(11): 113703–1–12 Ashely JC (1990) Energy loss rate and inelastic mean free path of low-energy electrons and positrons in condensed matter. J Electron Spectrosc Relat Phenom 50:323–334 Ashely JC, Anderson VE (1981) Interaction of low-energy electrons with silicon dioxide. J Electron Spectrosc Relat Phenom 24:127–148 Ashley JC (1988) Interaction of low-energy electrons with condensed matter: stopping powers and inelastic mean free paths from optical data. J Electron Spectrosc Relat Phenom 46:199–214 Berovic N, Pratontep S, Bryant A, Montouris A, Green RG (2002) The kinetics of radiation damage to the protein luciferase and recovery of enzyme activity after irradiation. Radiat Res 157(2):122–127 Bongeler R, Golla U, Ka¨ssens M, Reimer L, Schindler B, Senkel R, Spranck M (1993) Electron-specimen interactions in low-voltage scanning electron microscopy. Scanning 15(1):1–18
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Bousis C, Emfietzoglou D, Hadjidoukas P, Nikjoo H, Pathak A (2011) Monte Carlo calculations of low-energy electron dose-pointkernels in water using different stopping power approximations. Nucl Instrum Meth B 269(14):1650–1654 Brown JH (2006) Breaking symmetry in protein dimers: designs and functions. Protein Sci 15:1–13 Ding ZJ, Shimizu R (1996) A Monte Carlo modeling of electron interaction with solids including cascade secondary electron production. Scanning 18(2):92–113 Dingfelder M, Hantke D, Inokuti M, Paretzke HG (1998) Electron inelastic-scattering cross sections in liquid water. Radiat Phys Chem 53:1–18 Dingfelder M, Ritchie RH, Turner JE, Friedland W, Paretzke HG, Hamm RN (2008) Comparisons of calculations with PARTRAC and NOREC: transport of electrons in liquid water. Radiat Res 169(5):584–595 Emfietzoglou D (2003) Inelastic cross-sections for electron transport in liquid water: a comparison of dielectric models. Radiat Phys Chem 66(6):373–385 Emfietzoglou D, Nikjoo H (2005) The effect of model approximations on single-collision distributions of low-energy electrons in liquid water. Radiat Res 163(1):98–111 Emfietzoglou D, Karava K, Papamichael G, Moscovitch M (2003) Monte Carlo simulation of the energy loss of low-energy electrons in liquid water. Phys Med Biol 48(15):2355–2371 Emfietzoglou D, Cucinotta FA, Nikjoo H (2005) A complete dielectric model for liquid water: a solution of the Bethe ridge problem. Radiat Res 164(2):202–211 Emfietzoglou D, Kyriakou I, Abril I, Garcia-Molina R, Nikjoo H (2012a) Inelastic scattering of low-energy electrons in liquid water computed from optical-data models of the Bethe surface. Int J Radiat Biol 88(1–2):22–28 Emfietzoglou D, Kyriakou I, Garcia-Molina R, Abril I, Kostarelos K (2012b) Quasi first-principles Monte Carlo modeling of energy dissipation by low-energy electron beams in multi-walled carbon nanotube materials. Appl Phys Lett 100 (9):093113–1–5 Emfietzoglou D, Kyriakou I, Garcia-Molina R, Abril I (2013) The effect of static many-body local-field corrections to inelastic electron scattering in condensed media. J Appl Phys 114(14):144907–1–10 Ferna´ndez-Varea JM, Mayol R, Liljequist D, Salvat F (1993) Inelastic scattering of electrons in solids from a generalized oscillator strength model using optical and photoelectric data. J Phys Cond Matter 5:3593–3610 Ferna´ndez-Varea JM, Liljequist D, Csillag S, Ra¨ty R, Salvat F (1996) Monte Carlo simulation of 0.1–100 keV electron and positron transport in solids using optical data and partial wave methods. Nucl Instrum Meth B 108(1-2):35–50 Ferna´ndez-Varea JM, Llovet X, Salvat F (2005) Cross sections for electron interactions in condensed matter. Surf Interface Anal 37(11):824–832 Harmon JT, Nielsen TB, Kempner ES (1985) Molecular weight determinations from radiation inactivation. Meth Enzymol 117:65–94 Heintzman ND, Hon GC, Hawkins RD, Kheradpour P, Stark A, Harp LF, Ye Z, Lee LK, Stuart RK, Ching CW, Ching KA, Antosiewicz-Bourget JE, Liu H, Zhang X, Green RD, Lobanenkov VV, Stewart R, Thomson JA, Crawford GE, Kellis M, Ren B (2009) Histone modifications at human enhancers reflect global cell-type-specific gene expression. Nature 459:108–112 Jablonski A, Salvat F, Powell CJ (2003) NIST electron elastic scattering cross-section database-Version 3.1, National Institute of Standards and Technology, Gaithersburg, MD Joy DC, Joy CS, (1996) SEMATECH Report, TT#96063130 A-TR Kempner ES (2001) Effects of high-energy electrons and gamma rays directly on protein molecules. J Pharm Sci 90(10):1637–1646
Radiat Environ Biophys Kempner ES (2011) Direct effects of ionizing radiation on macromolecules. J Polym Sci B Polym Phys 49(12):827–831 Kotera M, Murata K, Nagami K (1981) Monte Carlo simulation of 1–10-keV electron scattering in an aluminum target. J Appl Phys 52(12):7403–7408 Kurniawan O, Ong VKS (2007) Investigation of range-energy relationships for low-energy electron beams in silicon and gallium nitride. Scanning 29(6):280–286 Lazaridis T, Karplus M (2000) Effective energy functions for protein structure prediction. Curr Opin Struct Biol 10(2):139–145 Lidzey DG, Berovic N, Chittock RS, Beynon TD, Wharton CW, Jckson JB, Parkinson NS (1995) A critical analysis of the use of radiation inactivation to measure the mass of protein. Radiat Res 143(2):181–186 Lowe ME, Kempner ES (1982) Radiation inactivation of the glycoprotein, invertase. J Biol Chem 257(21):12478–12480 Murata K (1974) Spatial distribution of backscattered electrons in the scanning electron microscopy and electron microprobe. J Appl Phys 45(9):4110–4117 Murata K, Kyser FD, Ting CH (1981) Monte Carlo simulation of fast secondary electron production in electron beam resists. J Appl Phys 52(7):4396–4405 Nave C (1995) Radiation damage in protein crystallography. Radiat Phys Chem 45(3):483–490 Nikjoo H (2003) Radiation track and DNA damage. Iran J Radiat Res 1(1):3–16 Nikjoo H, Goodhead DT, Charlton DE, Paretzke HG (1989) Energy deposition in small cylindrical targets by ultrasoft x-rays. Phys Med Biol 34(6):691–705 Nikjoo H, Goodhead DT, Charlton DE, Paretzke HG (1991) Energy deposition in small cylindrical targets by monoenergetic electrons. Int J Rad Biol 60(5):739–756 Nikjoo H, Uehara S, Emfietzoglou D, Cucinotta FA (2006) Trackstructure codes in radiation research. Radiat Meas 41(9–10):1052–1074 Nikjoo H, Emfietzoglou D, Watanabe R, Uehara S (2008) Can Monte Carlo track structure codes reveal reaction mechanism in DNA damage and improve radiation therapy. Radiat Phys Chem 77(10–12):1270–1279
Ochkur VI (1965) Ionization of the hydrogen atom by electron impact with allowance for the exchange. Sov Phys JETP 20(5):1175–1178 O’Neill P, Stevens DL, Garman EF (2002) Physical and chemical considerations of damage induced in protein crystals by synchrotron radiation: a radiation chemical perspective. J Synchrotron Rad 9(6):329–332 ¨ ztu¨rk N, Williamson W (1993) Monte Carlo simulation of keV O electron transport in solid media. J Appl Phys 74(7):4723–4728 Penn DR (1987) Electron mean-free-path calculations using a model dielectric function. Phys Rev B 35(2):482–486 Rauth AM, Simpson JA (1964) The energy loss of electrons in solids. Radiat Res 22(4):643–661 Ronan RS, Heinz WF, Kempner ES (1996) High-energy electron irradiation of protein and nucleic acids: collisional stopping power and average energy loss. Radiat Environ Biophys 35(3):159–162 Tan ZY, Xia YY, Zhao MW, Liu XD, Li F, Huang BD, Ji YJ (2004a) Electron stopping power and mean free path in organic compounds over the energy range of 20–10,000 eV. Nucl Instrum Meth B 222(1–2):27–43 Tan ZY, Xia YY, Zhao MW, Liu XD, Li F, Ji YJ, Huang BD (2004b) Cross sections of electron inelastic interactions in DNA. Radiat Environ Biophys 43(3):173–182 Tan ZY, Xia YY, Zhao MW, Liu XD (2006) Electron stopping power and inelastic mean free path in amino acids and protein over the energy range of 20–20,000 eV. Radiat Environ Biophys 45(2):135–143 Tan ZY, Dong L, Tang F (2012) Monte Carlo calculations of characteristic quantities of low-energy electron irradiation to spacecraft dielectrics. Nucl Instrum Meth B 285:86–93 Terwilliger TC, Waldo G, Peat TS, Newman JM, Chu K, Berendzen J (1998) Class-directed structure determination: foundation for a Protein structure initiative. Protein Sci 7:1851–1856 Uehara S, Nikjoo H, Goodhead DT (1993) Cross-sections for water vapour for the Monte Carlo electron track structure code from 10 eV to the MeV region. Phys Med Biol 37(12):1841–1858 van Attikum H, Gasser SM (2009) Crosstalk between histone modifications during the DNA damage response. Trends Cell Biol 19(5):207–217
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ORIGINAL PAPER
Monte Carlo calculations of energy deposition distributions of electrons below 20 keV in protein Zhenyu Tan • Wei Liu
Received: 4 October 2013 / Accepted: 26 January 2014 Ó Springer-Verlag Berlin Heidelberg 2014
Abstract The distributions of energy depositions of electrons in semi-infinite bulk protein and the radial dose distributions of point-isotropic mono-energetic electron sources [i.e., the so-called dose point kernel (DPK)] in protein have been systematically calculated in the energy range below 20 keV, based on Monte Carlo methods. The ranges of electrons have been evaluated by extrapolating two calculated distributions, respectively, and the evaluated ranges of electrons are compared with the electron mean path length in protein which has been calculated by using electron inelastic cross sections described in this work in the continuous-slowing-down approximation. It has been found that for a given energy, the electron mean path length is smaller than the electron range evaluated from DPK, but it is large compared to the electron range obtained from the energy deposition distributions of electrons in semi-infinite bulk protein. The energy dependences of the extrapolated electron ranges based on the two investigated distributions are given, respectively, in a power-law form. In addition, the DPK in protein has also been compared with that in liquid water. An evident difference between the two DPKs is observed. The calculations presented in this work may be useful in studies of radiation effects on proteins. Keywords Energy deposition Electron range Protein Monte Carlo simulation Radiation effect
Z. Tan (&) W. Liu School of Electrical Engineering, Shandong University, Jinan 250061, Shandong, People’s Republic of China e-mail: [email protected]
Introduction Proteins are very important bioorganic compounds. Histones, a kind of small molecular alkaline protein, are crucial components of chromatin and play an important role in cell differentiation and replication. Especially, it has been shown that histone modifications can regulate the cellular differentiation (Heintzman et al. 2009), and specific combinations of histone modifications can also affect the accumulation and function of DNA damage repair factors (van Attikum and Gasser 2009). Therefore, the functions and physical structures of proteins have been a subject of long-lasting experimental and theoretical interest (Terwilliger et al. 1998; Lazaridis and Karplus 2000; Brown 2006; Heintzman et al. 2009). On the other hand, radiation studies on bio-molecules have also been done for a long time and are of main concern of proteins or DNA (Kempner 2011). In these studies, the energy deposition and penetration range of energetic electrons in protein are two important quantities that require evaluation. Biological inactivity and main chain scission are the principal radiation effects in proteins reported (Lidzey et al. 1995; Ronan et al. 1996; Kempner 2001, 2011; Berovic et al. 2002). Radiation inactivation, where a single ionization of the protein is sufficient to inactivate it, can be utilized to directly determine the mass associated with the function of a protein (Ronan et al. 1996). For this determination, the knowledge of the average energy deposited by each inelastic event is required. In order to identify the relevant physical, chemical and biological parameters responsible for a radiation effect at the molecular level, the energy depositions in related biological targets, such as DNA, nucleosome, and chromatin fiber, have been calculated, in an effort to seek correlations between local energy depositions and the observed biological effectiveness of
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different radiation qualities (Nikjoo et al. 1989, 1991, 2003, 2008). These calculations, except those for DNA, involve the energy deposition of radiation in proteins because nucleosome and thus chromatin are composed of proteins and DNA. So far, radiation effects on protein molecules have been studied using high-energy electrons above 20 keV (Rauth and Simpson 1964; Lowe and Kempner 1982; Harmon et al. 1985; Kempner 2001), and the interactions and ranges of electrons in protein have also been investigated. In studies of the damage induced in protein crystals by synchrotron radiation, the energy of the emitted secondary electrons is higher than 10 keV; these electrons interact further with protein crystal generating a large number of new secondary electrons with sub-keV energies. Due to this, the penetration depth of these secondary electrons in proteins needs to be estimated (Nave 1995; O’Neill et al. 2002). In the present work, both the distributions of energy depositions of electrons in semi-infinite bulk protein and the radial dose distributions of point-isotropic mono-energetic electron sources [i.e., the so-called dose point kernel (DPK)] in protein are calculated, in the energy range below 20 keV. The calculations are based on a Monte Carlo method described in a previous work (Tan et al. 2012). Here, the ranges of electrons are evaluated by extrapolating two distributions, respectively, and compared with the range calculated using inelastic cross sections for electrons in the continuous-slowing-down approximation (CSDA). The energy dependence of the extrapolated electron ranges is presented in the power-law form. In addition, the DPK in protein is compared with that in liquid water.
Materials and methods In the present simulations, the interactions of energetic electrons with an organic medium include mainly two types of basic interaction, i.e., elastic and inelastic scattering. Elastic scattering is related to the interaction with the nuclei of target atoms, while inelastic scattering results from the interaction with the electronic shells of target atoms. The model used in the present simulations was described in more detail previously (Tan et al. 2012). Here, only a brief outline of this model is given. Monte Carlo model: elastic scattering The Mott cross section based on the partial wave method has been widely applied in Monte Carlo simulations of low-energy electron transport in various media. In the present work, the numerical Mott differential cross sections from the best known NIST database (Jablonski et al. 2003) for the elastic scattering of electrons are used. The Mott
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differential cross section derived from the relativistic wave equation of Dirac can be expressed as dr ¼ jf ðhÞj2 þjgðhÞj2 ; dX
ð1Þ
with f ðhÞ ¼
1 1 X fðl þ 1Þ½expð2idl Þ 1 þ l½expð2idl1 Þ 1g 2ik l¼0
pl ðcos hÞ; gðhÞ ¼
1 1 X ½expð2idl1 Þ expð2idl Þp1l ðcos hÞ; 2ik l¼1
ð2Þ ð3Þ
where h is the elastic scattering angle, dl is the phase shift corresponding to the lth partial wave, hk is the electron momentum, and pl(cosh) and p1l ðcoshÞ are the ordinary and associated Legendre polynomials, respectively.For lowenergy electron transport in organic media, the elastic interactions of electrons with the atoms of the different types of elements composing the organic compound of interest must be simulated frequently. This costs a large amount of cpu time, and therefore, the mean elastic cross section of an electron interacting with an atom is used here to calculate the elastic scattering, to allow for a high simulation efficiency. The mean differential elastic cross section for each atom is given by dre X ni dri ¼ ; dX n dX i
ð4Þ
P where ni is the atom density, n ¼ i ni , and dri/dX is the Mott differential cross section corresponding to the ith element composing the organic compound investigated.The mean total elastic cross section for each atom is then calculated via re ¼
Zp
dre 2p sin hdh: dX
ð5Þ
0
It is clear that using the mean cross section for each atom, the electron elastic scattering in a compound can be simulated in the same manner as that for a single element, obviously simplifying the calculation process of elastic scattering. It should be pointed out, however, that the described method of calculating electron elastic scattering neglects any condensed-phase effects because the NIST data are for isolated atoms only (Jablonski et al. 2003), and thus, the method may still be in error below an energy of a few hundred eV. A possible way to implement condensedphase effects in the electron elastic scattering can be found in Ferna´ndez-Varea et al. (2005). Moreover, the NIST data
Radiat Environ Biophys
allow the electron energy ranging from 50 eV to 300 keV. Hence, in the present simulations, the calculation of electron elastic scattering is limited by a cut-off energy of 50 eV. Monte Carlo model: inelastic scattering The dielectric response theory has been widely used for the description of electron inelastic scattering in the Monte Carlo simulation of electron transport in different media ¨ ztu¨rk and Williamson 1993; Ding and Shimizu 1996; (O Ferna´ndez-Varea et al. 1996; Emfietzoglou 2003; Nikjoo et al. 2006; Dingfelder et al. 2008; Akkerman et al. 2009). Previously, a dielectric model for simulating the inelastic interaction of electrons with organic molecules has been presented (Tan et al. 2012). In this model, the electron inelastic differential cross section is given by d2 rin 1 ¼ dð hxÞdq p a0 NE Z1 0 x 1 hq2 0 0 Im½1=eðx Þd x x Cex dx0 ; xq 2m 0
ð6Þ where E is the kinetic energy of the incident electron, a0 is the Bohr radius, N is the molecule density (or atom density), hx and hq are the energy loss and the momentum transfer, respectively, from the incident electron into the medium described by dielectric response function e(q,x), Im[-1/e(x)] is referred to as the optical energy-loss function (OELF), and Cex is given by 2 Q Q þ Cex ¼ 1 ; ð7Þ Eþ hx0 hx Eþ hx 0 hx with Q = h2q2/2 m, where m is the mass of the electron. Recently, Emfietzoglou et al. (2012a) presented a review of optical-data models describing low-energy electron inelastic scattering in a medium based on the dielectric response theory. In their review, different algorithms extending the optical data to finite momentum transfer are discussed, and it is shown that the models of Penn (1987) and Ashley (1988) represent a type of extension algorithm and overcome the need for a parametric fit of extendedDrude function by replacing the summation over a finite number of extended-Drude functions by an integration over a model energy-loss function. As shown previously (Tan et al. 2004a), Eq. 6 represents a simplification of the Penn model (1987) which is better associated with Ashley’s work (1988). In addition, in Eq. 6, Cex is a factor introduced to take into account the exchange effect between the incident electron and the electrons in the medium in the Ochkur
approximation (Ochkur 1965; Ferna´ndez-Varea et al. 1993, 2005). As pointed out by Ferna´ndez-Varea et al. (1993), the exchange effect correction in optical-data models was first considered by Ashely and Anderson (1981), who used a heuristic approach, i.e., a Mott-like scheme. After that, the Mott-like scheme has been used by many authors in their optical-data models to calculate the inelastic scattering of electrons in the media (Ashely 1988, 1990; Dingfelder et al. 1998; Akkerman and Akkerman 1999; Tan et al. 2004a; Emfietzoglou and Nikjoo 2005). More recently, the exchange–correlation effects in optical-data models of the dielectric function have been investigated in detail for the first time for carbon target (Emfietzoglou et al., 2013), where the different optical-data models corrected by the exchange–correlation were analyzed. The approach presented by Emfietzoglou et al. (2013) is manageable, and thus, the use of the exchange–correlation correction in the optical-data models for calculating the inelastic scattering of electrons in other target materials is expected in the future. For E0 = hx0 and DE = hx, integrating Eq. 6 and following Ferna´ndez-Varea et al. (1996), the differential cross sections with respect to resonance energy E0 and energy loss DE, respectively, are obtained, i.e., drin df ðE0 Þ ¼ rone ðE0 Þ dE0 dE0
ð8Þ
and drone ¼ dDE
ZQþ
p e4 1 dðDE E0 QÞCex dQ; E DEQ
ð9Þ
Q
pffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi with Q ¼ ð E E DEÞ2 , where rone(E0 ) is the cross section for exciting a unit-strength oscillator with resonance energy E0 . Equations 8 and 9 will be used for determining resonance energy E0 and energy loss DE by random sampling in the simulation of electron inelastic scattering. The detailed procedure of simulating electron transport in an organic medium based on the model given above can be found in Tan et al. (2012), where the model was justified through a series of calculations and comparisons with published experimental and theoretical results. Here, as an example of using this model, Fig. 1 presents (a) the calculated electron backscattering coefficients for bulk PMMA (C5H8O2) and (b) the calculated energy distributions of transmitted electrons of PMMA film. In Fig. 1a, the experiments of Joy and Joy (1996) and the theoretical results from Murata et al. (1981) are also plotted for comparison. Figure 1a shows that in the energy range of E0 B 20 keV, the present calculations are in satisfactory agreement with the experiment and with the theoretical results. Figure 1b compares the energy distributions of transmitted electrons for PMMA film of thickness
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Radiat Environ Biophys 0.16
10
-13
protein σe σin
(a) PMMA 2
cross section (cm )
This work
0.12
Joy and Joy (1996)
ηB
Murata et al. (1981)
0.08
10
-14
10
-15
0.04
0
5
10
15
20
E0 (keV)
10
2
10
3
10
4
energy (eV)
16
Fig. 2 Total elastic and inelastic cross sections of electrons for an equivalent unit (C107H197N29O49S2)n of the considered protein as a function of energy
(b) PMMA, 20 keV Adesida et al. (1980)
dηT /dU
12
This work
8
4
0 0.5
1600 nm
0.6
0.7
0.8
0.9
1.0
U=E/E0 Fig. 1 Electron backscattering coefficient gB as a function of primary energy E0 for bulk PMMA (a); Energy distributions of transmitted electrons for PMMA film with thickness 1,600 nm at E0 = 20 keV (b)
1,600 nm at 20 keV from the present calculation with those from Adesida et al. (1980). The comparison indicates reasonable agreement between the two theoretical results.
Results and discussion Cross sections In the present work, the considered protein is the same as that used by Tan et al. (2006), with an equivalent unit (C107H197N29O49S2)n and a mass density of 1.3 g/cm3. For the considered protein, there is no available experimental optical data, and thus, its OELF was evaluated here with the use of the empirical approach (Tan et al. 2004a, b).
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Using the evaluated OELF, pre-calculated look-up tables were calculated for determining resonance energy E0 and energy loss DE. In addition, in the simulations, the number of simulated tracks was 2 9 105–59105 for each primary energy, and the cutoff energy of terminating a track was taken as 50 eV because of the limitation of the minimum energy of 50 eV for the elastic cross-sectional calculations, as stated above. The total elastic and inelastic cross sections describe the probability with which an incident electron interacting with the medium may be scattered elastically or inelastically. Figure 2 shows the resulting calculated total elastic and inelastic cross sections of electrons for an equivalent unit of the considered protein as a function of electron energy in the range below 20 keV. It can be seen from Fig. 2 that above about 200 eV the probability for inelastic scattering of an incident electron is larger than that for elastic scattering, while the reverse is true for energies below 200 eV. Energy deposition in semi-infinite bulk protein Using the Monte Carlo model given above, the depth distributions of the energy deposition for electrons with primary energies lower than 20 keV in semi-infinite bulk protein were calculated. The distributions are described by dE(z)/dz, where z is the depth into the protein. Figure 3 presents these distributions for various primary energies E0. It is clearly observed from Fig. 3 that a larger primary energy results in a larger penetration range, a deeper position of the peak, and a smaller value of the peak. The range of electrons which penetrate into a semiinfinite sample is a quantity of interest for practical
Radiat Environ Biophys 40
16
1 keV
protein
protein, E0=5 keV 12
dE/dZ (eV/nm)
dE/dZ (eV/nm)
30
3 keV
20
5 keV
10
8
ED
Rext
4
10 keV 20 keV
0 0 10
1
10
2
10
3
10
0
4
10
200
400
depth z (nm)
600
800
depth z (nm)
1:5 RED text ¼ 40:82E0
for 0:5keV E0 \1:5 keV;
ð10Þ
1:72 RED ext ¼ 37:01E0
for 1:5keV E0 20 keV;
ð11Þ
Using Eqs. 10 and 11, RED ext can be evaluated conveniently. For the determination of RED ext s, because the models for elastic and inelastic scattering used in the present
4
10
protein
3
10
ED
applications (Kotera et al. 1981; Bongeler et al. 1993; Akkerman et al. 2009). Specifically, the range of the electrons penetrating into protein is of importance for exploring the radiation damage to protein (Nave 1995; Kempner 2001; O’Neill et al. 2002). The range of electrons is characterized by the quantity R, for which there are different definitions. A detailed description of these different definitions can be found elsewhere (Akkerman et al. 2009; Emfietzoglou et al. 2012b). A usual definition of R is the so-called extrapolated range Rext (Akkerman et al. 2009). The extrapolated range can be determined by extrapolating the linear part of the depth distribution of the calculated energy deposition, dE(z)/dz, to the z-axis, as illustrated in Fig. 4, which presents dE(z)/dz as a function of depth z at E0 = 5 keV (also presented in Fig. 3). In this figure, the extrapolated range is denoted by RED ext . Employing this approach, obtained RED values as a funcext tion of primary energy E0 are shown in Fig. 5. The fact that the range of electron obey approximately the power law R ¼ AE0B , where A and B are two coefficients, has already been shown for other solids by experiments and theoretical calculations (Kotera et al. 1981; Bongeler et al. 1993; Kurniawan and Ong 2007; Akkerman et al. 2009). For the bulk protein investigated here, Fig. 5 demonstrates that the E0 dependence of RED ext (in unit nm) can be fitted by
Fig. 4 Depth distribution of energy deposition, dE(z)/dz, at E0 = 5 keV for bulk protein, and RED ext determined by extrapolating the linear part of dE(z)/dz to the z-axis
Rext (nm)
Fig. 3 Depth distributions of energy deposition in bulk protein for different primary energies E0
2
10
1
10
1
10
E0 (keV) Fig. 5 Extrapolated electron range RED ext energy E0 for bulk protein
as a function of primary
simulations may include errors at energies below 500 eV, the RED ext s for E0 \ 500 eV have not been considered. Note that, because the RED ext values did not follow a clear linear relationship in the log10 RED ext versus log10 E0 representation, for the whole energy range, the E0 dependence of RED ext was fitted separately for two energy regions. Dose point kernel calculation The radial dose distribution of point-isotropic mono-energetic electron sources, i.e., the DPK, in liquid water is
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Radiat Environ Biophys 10
0 4
10
protein
protein 0.5 keV
-1
10
DPK
Rext
3
Range (nm)
DPK
1 keV -2
10
3 keV 5 keV
10
2
10
-3
10
1
10
-4
10
0
1
10
10
2
10
3
10
1
radial distance (nm)
E0 (keV)
Fig. 6 Dose point kernel (DPK) profiles for different primary energies in protein
usually calculated for understanding the interaction between ionizing radiation and a biological medium. In the calculation, the starting point of each electron track is placed at the origin of coordinates, around which liquid water is divided into a series of spherical shells at Dr intervals from the origin, and then DPK is given by DPKðE0 ; rÞ ¼
DEðrÞ ; Ntr E0
ð12Þ
where Ntr is the number of simulated tracks and DE(r) is the deposited energy due to Ntr primary electrons in a concentric spherical shell with radius r and thickness Dr. Based on Eq. 12, DPK calculations for liquid water were performed using different Monte Carlo codes (Uehara et al. 1993; Emfietzoglou et al. 2003; Bousis et al. 2011). In this work, the DPK values in protein were calculated for energies below 20 keV. Figure 6 presents the calculated DPK values for selected values of E0, where Dr is set to 1 nm and Ntr is taken as 2 9 105 for 0.5 keV, 1 keV, and 3 keV, and as 5 9 105 for 5 keV. From Fig. 6, it is clear that the primary energy dependence of the DPK is similar to that of dE(z)/dz shown in Fig. 3. In practice, DPK represents the radial distribution of energy deposition of energetic electrons in the medium, and therefore, it is also called the fraction of absorbed energy (Uehara et al. 1993; Emfietzoglou et al. 2003). Due to this, and in a manner analogous to that indicated in Fig. 4 for dE(z)/dz, one can obtain an extrapolated range in the radial direction through the DPK for a given E0. Here, this range is denoted DPK by RDPK ext . Figure 7 gives Rext as a function of primary energy E0. Similarly as above, the primary energy dependence of RDPK ext can also be fitted by
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10
Fig. 7 Extrapolated electron range RDPK ext from DPK and the calculated penetration parameter \V[ in protein as a function of primary energy E0 1:51 RDPK ext ¼ 47:45E0 1:7 RDPK ext ¼ 42:85E0
for 0:5 keV E0 \1:5 keV; for 1:5 keV E0 20 keV;
ð13Þ ð14Þ
In this work, we have also simulated the penetration parameter \V[ of low-energy electrons in protein, where V is defined as the distance of the most distant electron transfer point relative to the starting point, and \V[ is the mean value of V. The calculated \V[ values are also displayed in Fig. 7 for comparison. It can be seen that\V[ is smaller than RDPK ext for a given E0. Another well-known definition of the electron range R is the electron mean path length. According to this definition, the electron range is usually indicated as RCSDA, and it is calculated in the CSDA by RCSDA ¼
ZE0
ðdE=dxÞ1 dE;
ð15Þ
Ecut
where -dE/dx is the collision stopping power (SP), and Ecut is the energy cutoff in the calculation of RCSDA . In this work, using -dE/dx based on the dielectric model described by Eq. 6 and taking Ecut as 50 eV, RCSDA was calculated in the energy range below 20 keV and DPK compared with RED ext and Rext , as shown in Fig. 8. Interestingly, from this comparison, it can be observed that DPK RCSDA is larger than RED ext , but smaller than Rext . The reason for RCSDA \RDPK ext may be attributed to the following: In the calculation of DPK, all the electrons starting from their point of origin contribute their energies in total to DPK. On the other hand, RCSDA is only a mean path length traveled
Radiat Environ Biophys 0.05
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protein
protein liquid water: using empirical OELF using Emfietzoglou's OELF
0.04
103
DPK
Rext
ED
0.03
Rext
E0=1 keV
DPK
Range (nm)
RCSDA
0.02
102
0.01
101 0.00
100
0
101
by an energetic electron, and thus, it becomes small, ED compared to RDPK ext . In addition, RCSDA [ Rext is partly due to the fact that the contribution of the backscattered electrons to the linear part of dE(z)/dz is very small (Murata 1974). Water is usually used as a substitute of biological tissues in radiation biology (Nikjoo et al. 2006). Therefore, it is of significance to compare the DPK in protein with that in water. Employing the Monte Carlo model described here, DPK for liquid water was calculated. Figure 9 presents a comparison between DPKs in liquid water and in protein, respectively, at 1 keV, where the DPKs in liquid water were calculated by the use of the OELFs from the empirical evaluation (Tan et al. 2004a, b) and from the parameterized model of Emfietzoglou et al. (2005), respectively. It can be seen from Fig. 9 that the distribution range of the DPK in protein is obviously small, compared to that in liquid water. Therefore, when liquid water is used as a substitute for protein in calculations of energy depositions, this difference should be taken into account. In addition, the difference between the DPKs in liquid water calculated by using the OELFs from Tan et al. (2004a, b) and from Emfietzoglou et al. (2005), respectively, is very small. Actually, the distribution range of the DPK is mainly determined by the SP of the medium for electrons. A large SP results in a small distribution range. For the compounds, the non-relativistic Bethe formula describing the SP (in eV/ nm) can be written as (Bousis et al. 2011) dE qZ 1:166E ¼ 7850 ln ; ð16Þ dx ME I
40
60
80
radial distance (nm)
E0 (keV) Fig. 8 Comparison between different electron ranges in protein, i.e., DPK RCSDA, RED ext ; and Rext
20
Fig. 9 Comparison between DPKs in protein and liquid water, respectively. OELF optical energy-loss function; dotted line calculated with an OELF from Tan et al. (2004a, b); dashed line calculated with an OELF from Emfietzoglou et al. (2005)
where E is the electron energy (in eV), M is the molecular weight of the medium with Z electrons per molecule, q is the mean excitation the mass density (in g/cm3), and Iis energy (in eV). Using the evaluated OELF for protein and liquid water, are 71.4 eV and 72.5 eV, respectively the calculated Is (Tan et al. 2004a, 2006), being very close to each other. In addition, the ratio of (Z/M)protein using (C107H197N29O49S2)n to (Z/M)water is about 0.95. Due to this, the difference between the distribution ranges of the DPKs for protein and liquid water is mainly induced by their different mass densities (1.3 g/cm3 for protein and 1 g/cm3 for liquid water). In a previous work (Tan et al. 2006), the SPs of protein and DNA for electrons in the energy range below 10 keV were calculated, and it was shown that these are systematically smaller for DNA than for protein. Therefore, it can be expected that the distribution range of the DPK for electrons below 10 keV in DNA is larger than that in protein. However, it is still required to calculate in detail the distributions of the DPKs of low-energy electrons in DNA in future studies.
Conclusion In this work, systematic Monte Carlo calculations were performed in an effort to study energy deposition distributions of electrons in semi-infinite bulk protein and radial dose distributions of point-isotropic mono-energetic electron sources (DPK) in protein, for electron energies below
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20 keV. Evaluation of electron ranges was done by extrapolating the two calculated distributions, respectively. Electron mean path lengths in protein were also calculated by using electron inelastic cross sections in the CSDA and compared with the electron ranges evaluated from the two energy deposition distributions. Furthermore, using the present Monte Carlo method, the DPK in liquid water was calculated and compared with that in protein. The present work showed that in the considered energy range, the electron mean path length is about 2–8 % smaller than that evaluated from DPK, but it is about 6–7 % larger than that obtained from the energy deposition distributions of electrons in semi-infinite bulk protein. These results may provide a reference when estimating electron ranges in studies of radiation effects on proteins. In addition, energy dependences of the extrapolated electron ranges based on the two distributions were described in a power-law form for conveniently evaluating the electron range in proteins in future studies. Especially, the present calculations indicated that the distribution range of the DPK in protein is obviously small in contrast with that in liquid water, due mainly to their different mass densities. This difference should be taken into account when liquid water is used as a substitute for protein in calculations of energy depositions. Acknowledgments The authors are grateful to Drs. D. C. Joy and C. S. Joy of the University of Tennessee for providing their database of the compiled experimental data of electron backscattered coefficients. This work was supported by the Foundation of Ministry of Education of China under Grant No.20120131110012.
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