Applying Petri nets to modeling the chemical stage of radiobiological mechanism

Journal of Physics and Chemistry of Solids 78 (2015) 127–136

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Applying Petri nets to modeling the chemical stage of radiobiological mechanism J. Barilla a,n, M. Lokajíček b, H. Pisaková b, P. Simr a a b

J. E. Purkinje University in Usti nad Labem, Faculty of Science, České mládeže 8, 400 96 Ústi nad Labem, Czech Republic Institute of Physics, v.v.i., Academy of Sciences of the Czech Republic, Na Slovance 2, 182 21 Praha 8, Czech Republic

art ic l e i nf o

a b s t r a c t

Article history: Received 31 July 2014 Received in revised form 20 November 2014 Accepted 25 November 2014 Available online 27 November 2014

The chemical stage represents important part of radiological mechanism as double strand breaks of DNA molecules represent main damages leading to final biological effect. These breaks are formed mainly by water radicals arising in clusters formed by densely ionizing ends of primary or secondary charged particles in neighborhood of a DNA molecule. The given effect may be significantly influenced by other species present in water, which may depend on the size and diffusion of corresponding clusters. We have already proposed a model describing the corresponding process (i.e., the combined effect of cluster diffusion and chemical reactions) running in individual radical clusters and influencing the formation probability of main damages (i.e., DSBs). Now a full number of corresponding species will be considered. With the help of Continuous Petri nets it will then be possible to follow the time evolution of corresponding species in individual clusters, which might be important especially in the case of studying the biological effect of very low-LET radiation. The results in deoxygenated water will be presented; the ratio of final and initial contents of corresponding species being in good agreement with values established experimentally. & 2014 Elsevier Ltd. All rights reserved.

Keywords: Radiation damage Organic compounds Critical phenomena Defects

1. Introduction Any ionizing particle incident into a biological object (and having sufficient energy) may have significant influence on the life of a corresponding cell. The basic effect may be represented by the formation of SSB or DSB (single or double strand breaks) in individual DNA molecules. The whole radiobiological mechanism consists then always of three stages: physical, chemical and biological. While the physical stage (corresponding to energy transfer to irradiated medium) is known quite well the actual characteristics of two other stages often influencing the corresponding biological effect very substantially are still rather open. As in water medium the direct effect of incident particles may be practically fully neglected the chemical stage always represents important part of the corresponding mechanism. The given damages in DNA molecules are caused by radicals arising in clusters from water molecules by densely ionizing track ends of individual ionizing particles in the neighborhood of individual molecules. Especially in the case of low-LET radiation the final biological effect may be rather strongly influenced by the presence of some other species in corresponding water medium; oxygen being one of the most important species. n

Corresponding author. E-mail address: [email protected] (J. Barilla).

http://dx.doi.org/10.1016/j.jpcs.2014.11.016 0022-3697/& 2014 Elsevier Ltd. All rights reserved.

The individual radicals present in a cluster may always form one SSB that may be practically quickly repaired in individual DNA molecules. More efficient DSB may be also repaired in diploid cells but the repair time is much greater. Consequently, at greater radiation doses a greater number of these DSBs may be formed in individual DNA molecules, which may lead to destroying corresponding molecules and eventually to even killing of diploid cells. The number of efficient radicals in individual clusters may then be influenced (lowered or heightened) by the presence of other species in corresponding water medium. Then the efficiency of DSB formation lowers quickly by cluster diffusion. Great attention has been devoted to estimating the initial yields of radicals and ions (e−aq , H•, OH• , H3 O+ ) and associated products (OH−, H2, H2O2, O−2 , HO•2) in the time when the clusters have been formed; see different compilations concerning the radiation chemistry of water (e.g. [10,11,15–17,19,24–26,8]). For low-LET radiation the radical clusters of different sizes may be formed according to the energy of secondary electrons that are responsible for cluster formation. According to Mozumder and Magee [18] the excitation energy values of efficient clusters lie in the interval from several tens of eV to 500 eV. In the radiation chemistry research the average number corresponding to different energy values has been established. However, it is possible to say that until now the efficiency of different clusters in DSB formation and of different radiomodifiers (in final biological effect) has not been sufficiently evaluated.

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J. Barilla et al. / Journal of Physics and Chemistry of Solids 78 (2015) 127–136

We have started to study the corresponding problem already earlier, the contemporary result of chemical reactions and cluster diffusion having been described with the help of corresponding differential equations (see [3–7]). The average excitation energy of efficient clusters (in the case of Co60 radiation [6]) should be approximately 300 eV (estimated according to results obtained in radiation chemistry studies). We should ask, however, how the efficiency of corresponding individual clusters changes in dependence on their sizes. And further, whether the distribution of radicals in all clusters corresponds to average size when the radicals are formed in diverse processes requiring different excitation energy values? Or how does the corresponding efficiency evolve during cluster diffusion if the interaction with DNA molecules occurs some time after cluster formation? These questions may be studied with the help of Continuous Petri nets that have been introduced by us recently [7]. It has been demonstrated in the case of 100 eV clusters arising in deoxygenated water that the average yields of individual radicals in the time of cluster formation and at the end of cluster diffusion correspond well to the evolution values described with the help of Petri nets. In this paper we shall try to derive some further characteristics that might contribute to the solution of the mentioned problems. In Section 2 the influence of changing concentrations due to cluster diffusion dependence on chemical reactions running in individual clusters will be derived. The system of all processes running in clusters will be represented with the help of Continuous Petri nets in Section 3. Main characteristics provided with the help of Petri nets and the comparison with experimental data will be presented in Section 4. Some concluding remarks will be introduced in Section 5.

2. Mathematical model of processes running in diffusing clusters The corresponding mathematical model of chemical stage has started from the assumption that the given process is mediated by diffusing radical clusters containing non-homogeneous concentrations of individual species [19]. And macroscopic laws have been used to describe the diffusion of radiation-induced objects and concentration changes caused by different chemical reactions. In developing the given mathematical model we have come out of the prescribed diffusion model of the radiation track proposed by Schwartz [21]. In contrast to the solution published by Schwarz, we have converted our mathematical model to the solution of ordinary differential equations, which has significantly accelerated the simulation process. To establish the influence of the cluster diffusion (on corresponding concentration changes) we shall assume for simplicity that this diffusion goes in a limited spherical volume [19]. The time dependence of the diffusion process may then be described with the help of partial differential equations:

∂ci 1 ∂ ⎛ 2 ∂ci ⎞ ⎜r ⎟, = Di ∂t r 2 ∂r ⎝ ∂r ⎠

(1)

where ci (r , t) represent the time dependences of concentration distributions of individual species i and r denoting the distance from the cluster center. By solving this equation one obtains

⎛ ⎞ Ni r2 ⎟ ci (r , t) = exp ⎜− , ⎜ ⎟ 8 (πDi t)3 ⎝ 4D i t ⎠

(2)

where Ni is the (initial) number of species i in the cluster, Di are corresponding diffusion coefficients and t is the time from the

start of the diffusion in the theoretical center of the given cluster. The average cluster radius then equals

r¯i (t) =

1 Ni

∫0

∞

rci (r , t)4πr 2 dr

(3)

or

r¯i (t) =

π 2 (πDt)3

∫0

∞

⎛ ⎞ r2 ⎟ r 3 exp ⎜⎜− ⎟ dr , ⎝ 4Dt ⎠

(4)

and the time dependence of the cluster volume may then be defined as

Vi (t) =

4 3 πr¯i (t) 3

(5)

or

Vi (t) =

⎛ D 3t 3 ⎞ ⎜⎜ i ⎟⎟ . ⎝ π ⎠

256 3

(6)

One can see from the last equation that the volumes occupied by individual diffusing species at a given time are practically independent of the number of corresponding species, depending only on their diffusion coefficients. Differentiating Eq. (6) we obtain

⎛ D 3t ⎞ dVi (t) = 128 ⎜⎜ i ⎟⎟ . dt ⎝ π ⎠

(7)

The given formulas (6) and (7) may be accepted as sufficient for characterizing the influence of diffusion process at least in the first approximation (even if the cluster diffusion will not go in fully spherically symmetrical way). As the size of cluster is small (about tens of nm), the concentration profile of radicals in clusters may be assumed to be Gaussian, and it is then possible to introduce average cluster concentrations c¯i (t) of individual species:

c¯i (t) =

Ni (t) , Vi (t)

(8)

where Ni(t) is the number of corresponding chemical species in a cluster in a given instant and Vi(t) describes the time dependence of cluster volume established according to Eq. (6), which may be regarded as sufficient approximation for simulating the dynamics of chemical processes in diffusing clusters. The average concentration change of individual chemical substances at any time may then be then given as the sum of two processes (diffusion þ chemical reactions). One can write

dc¯i (t) c¯i dVi =− − dt Vi dt

∑ kij c¯i (t) c¯ j (t) + ∑

(i) ¯ k jk c j (t) c¯ k (t).

j, k ≠ i

j

(9)

The first term of the right-hand side of the system of ordinary differential equations (9) represents the diffusive contribution to the evolution of ci, while the second and third terms represent removal and production of species i by chemical reactions, respectively. Using Eq. (8) one obtains the system of ordinary differential equations:

dNi (t) =− dt

∑ kij j

Ni (t) N j (t) V j (t)

+ Vi (t)

∑ j, k ≠ i

k (jki)

N j (t) Nk (t) V j (t) Vk (t)

, (10)

Eq. (10) together with Eq. (7) describes the dynamics of chemical reactions at any time t when the influence of cluster diffusion is respected. Chemical reactions are influenced by diffusion processes causing a concentration change of individual particles

J. Barilla et al. / Journal of Physics and Chemistry of Solids 78 (2015) 127–136

due to the time change of cluster volume according to the differential equation (7). It has been assumed that the diffusion of cluster has been determined in principle by diffusion coefficients of main species being present in it. The concentrations of these species may change, of course, on the basis of the corresponding chemical reactions. Consequently, the efficiency of forming SSB or DSB may be different according to the instant when a cluster and a DNA molecule meet; the probability of direct meeting at the impact of ionizing particle is negligibly small for low-LET radiation, occurring due to cluster diffusion and temperature movement. The other problem concerns, of course, establishing the actual initial condition concerning proper chemical phase beginning. The whole diffusion process has had to be described as starting in a theoretical center of the corresponding cluster. However, the actual chemical phase starts at a time t0 after this theoretical beginning; corresponding to r¯0 (t0 ) or to V j (0) = 4/3πr¯03. It has been derived in [6] by fitting experimental data that in the case of Co60 radiation the size of clusters efficient in DSB formation has been approximately t0 ≅ 13 ns, which has corresponded to r¯0 ≅ 13.5 nm . The corresponding value will also be used in our approach. In determining the numbers of individual species we shall start from the initial number ratios of individual radicals determined already on the basis of experimental data in the given case (see e.g. [10,17]); only their actual numbers in efficient clusters will be established on the basis of corresponding experimental data.

3. Simulation of processes running in diffusing clusters using Continuous Petri nets We shall now simulate the dynamics of chemical processes (described by Eq. (10) with the help of Continuous Petri nets (see e.g. [14,22,23]). Petri nets are a graphical modeling tool for the formal description of the flow of activities in complex systems. They are particularly suited to represent logical interactions among different parts or activities in a system. Typical situations that can be modeled by Petri nets are synchronization, sequentiality, concurrency and conflict. For simulating the chemical stage in water radiolysis the Continuous Petri nets are very suitable because they enable us to model the time dependence of chemical reactions and diffusion processes simultaneously. Great advantage of Petri nets for modeling complicated distributed systems consists then in the possibility of being easily created with the help of graphical tools and of rapid analysis of the corresponding system which enables us to optimize the parameters of a mathematical model (see e.g. [13,14,22,23]). Unlike the classical methods creating mathematical models with the help of programing languages the Petri nets

129

enable us to form gradually a mathematical model via places and transitions. The model can be tested, easily expanded and improved at each stage. Continuous Petri nets consist of three main elements: places, transitions and arcs. The places represent the state of the system; see Fig. 1. Each place is marked by a real number, which determines the amount (value) of monitored parameter (e.g. the concentration of a chemical substance in a volume, etc.). The real number determining the state of a monitored place can be changed via transition. The dynamic change of a place is given by the corresponding transitions determined with the help of differential equations. Each place can be changed only via the transitions which are linked to it. The dynamic changes of all places caused by the corresponding transitions occur simultaneously. It is the process similar to solving a system of differential equations with the difference that here we do not have anything complicated to program and debug. Individual places and transitions are entered graphically and then joined using the arcs. The right side of the differential equation for each transition is given by a simple term. The simulation model is created by adding gradually places and transitions, and each step can be tested. Complex model with a large number of places and transitions can be created very quickly. The model allows rapid analysis when different free parameters are to be optimized according to corresponding data. The standard approach in modeling ambient water analysis consists usually of combining the deterministic diffusion-kinetic computations and the stochastic approach combining Monte Carlo simulation (see e.g. [10]). Monte Carlo simulation is being used with good results for modeling physical and physico-chemical stage. However, while for this initial stage this method is suitable it is not so for the proper chemical stage. This stage begins at the moment when the formation of a radical cluster is terminated, and mutual chemical reactions between the radicals and other substances begin to run with the simultaneous diffusion of radical clusters. In our approach we are taking the initial yields of radicals in a cluster as the input into our model. To simulate the proper chemical stage it is advantageous to use our analytical model because it provides a natural way to describe the dynamics of chemical reactions using reaction constants and diffusion coefficients known already from the literature. It allows us to perform fast analysis of the model; very complex models that include large number of chemical reactions and interactions with DNA can be simulated. In this paper we shall focus on extending the previously proposed model [7] to obtain better results concerning mainly the chemical stage; consequently, we shall use the experimental results obtained in the literature to verify our model and shall demonstrate mainly some new possibilities provided by the use of Petri nets.

T(i+j)

Vi

Ti m1

Ni

T(j+k) m2

Fig. 1. Simple Petri nets.

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J. Barilla et al. / Journal of Physics and Chemistry of Solids 78 (2015) 127–136

Fig. 2. Petri nets.

A Continuous Petri nets can be defined mathematically as quadruple N ¼ (P, T, Pre, Post) where:

 P is the set of places represented (see Fig. 1) by circles, |P | = m is number of all places; in our case m ¼45 (see Fig. 2);

 T is the set of transitions represented by bars or rectangles, |T | = n; n ¼56 represents all transitions shown in Fig. 2;

 Pre: P × T → N (Q ) is the pre-incidence function that specifies the arcs from places to transitions;

 Post: T × P → N (Q ) is the post-incidence function that specifies the arcs from transitions to places. In Petri nets individual species i (present in diffusing clusters)

will be represented by continuous places Pi, and volumes Vj by continuous places Pj. Each place Pi is marked by real number expressing number of species i (Ni), and each place Pj is marked by real number expressing the size of volume Vj of the diffusing cluster. Each place Pi can be changed only by connected transitions. The time change of the continuous place Pi is via continuous transitions Ti + j and T((ji)+ k) . According to Eqs. (10) and (7) we can express the time change of the number of species i (Ni) with the help of the set of continuous transitions Tj + k and T((ji)+ k) in form

dNi (t) = dt

∑ T(i + j) + ∑ j

j, k ≠ i

T((ji)+ k)

(11)

J. Barilla et al. / Journal of Physics and Chemistry of Solids 78 (2015) 127–136

dVi (t) = Ti dt

(12)

T(i + j) = − kij

=

Table 2 Recombination reactions [10,12]. Rate constants (dm3 mole  1 s  1)

Reactions

where

T((ji)+ k)

131

Ni (t) N j (t) V j (t)

(i) Vi (t) k jk

N j (t) Nk (t) V j (t) Vk (t)

(13) .

(14)

The transition (13) expresses the speed of the number decrease of species i (Ni) due to reaction of species i with species j and the transition (14) expresses the speed of the number increase of species i (Ni) due to reaction of species j with species k for j, k ≠ i . It is, of course, necessary to use the value of volume Vi, which corresponds to individual species i as the result of diffusion into the surrounding. The increase of this volume occurs via continuous transitions Ti according to Eq. (7):

⎛ D 3t ⎞ Ti = 128 ⎜⎜ i ⎟⎟ . ⎝ π ⎠

(15)

It means that each continuous transition Ti, T(i + j) , T((ji)+ k) causes time change of place to which it is pointed. This changes run according to differential equations (10) and (7). All changes take place simultaneously via continuous transitions Ti and influence each other. In the case, when the arc points from place (circle) to transition (rectangle) the place decreases according to Eq. (13) and in the case, when the arc points from transition to place the given place increases according to Eq. (14) (due to Ni) or according to Eq. (15) (due to Vi); see Fig. 1. We shall assume that the radicals shown in Table 1 may be involved in processes responsible for radiobiological effect (when oxygen is present); their diffusion coefficients taken from the literature having been introduced, too. As to the content of radicals in the clusters in deoxygenated case we shall assume that only the following ones will be present at t0: H•, OH•, e−aq and H3 O+ . Considered chemical reactions are then introduced in Table 2; the reaction rates (taken from the literature in corresponding units) are also given in this table (see [12]). To create the given mathematical model characterizing the evolution of the corresponding cluster after its formation with the help of Continuous Petri nets we have used the system Visual object net þ þ (see [20]). In this tool the transition functions may be included as well as the places that are not joined with it. It enables us to create simpler graphical simulation model. The whole process dynamics is expressed graphically by continuous Petri nets in Fig. 2, where the places corresponding to

Substance

Diffusion coefficient

H• + H• • e− aq + H

⟶ ⟶

3.

− e− aq + eaq

⟶

H2 + 2OH−

2.5 × 1010 6  109

4.

• e− aq + OH

⟶

OH− + H2 O

3  1010

5.

H• + OH•

⟶

H2 O

6.

OH• + OH•

⟶

H2 O2

2.4 × 1010 4  109

7.

H3O+ + e− aq

⟶

H• + H2 O

8.

HO•2 + H•

⟶

H2 O2

2.3 × 1010 1  1010

9.

HO•2 + OH•

⟶

H2 O + O2

1  1010

10.

HO•2 + HO•2

⟶

H2 O2 + O2

2  106

11.

O− 2

⟶

HO•2

3  1010

12.

H3O+ + OH−

⟶

H2 O

13.

H•

+ H2 O2

⟶

H2 O +

14.

e− aq + H2 O2

⟶

OH• + OH−

15.

OH• + H2 O2

⟶

H2 O + HO•2

1.2 × 1010 5  107

16.

OH• + H2

⟶

H2 O + H•

6  107

17.

HO•2

⟶

H3O+ + O− 2

1  106

+ H3

(nm2 ns−1)

Species amount

Designation of diff. coefficients

1. 2. 3.

H• OH• e− aq

7.0 2.2 4.9

NH NOH Ne

DH DOH De

4.

H3O+

9.5

N H3 O+

D H3 O+

5.

OH 

5.3

NOH −

6.

H2

5

NH2

DOH − D H2

7.

H2 O2

2.2

NH2 O2

D H2 O2

8.

O− 2

1.8

NO2−

DO2−

9.

HO•2

2.3

NHO2

D HO2

O+

H2 H2 + OH−

1  1011 1  108

OH•

individual radicals (specified in Table 1) are marked as H, OH, e, H3O, OHM, H2, H2O2, O2M, HO2. The other places correspond to the average volumes of individual radicals, the rates of chemical reactions, the diffusion coefficients and also to the value of π used in the corresponding expressions. A value is assigned to each place; it represents the amount of corresponding species. Individual places may be changed via corresponding transitions. Each transition represents the change rate of species number due to chemical reaction according to Table 2. All places and transitions are shown in Fig. 2. The places which are at the top of the figure represent constants from Tables 1 and 2 (diffusion coefficients and chemical rate values). The places at the left of Fig. 2 represent volumes, where the corresponding particles are located. Their changes take place through the connected transitions. Other places which represent amounts of individual species are changed due to transitions that are connected with them. The whole process may be described by a system of ordinary differential equations, which includes the influence of chemical reactions and diffusion of radicals simultaneously in the form

dNH NH (t) NH (t) NH (t) Ne (t) NH (t) NOH (t) = − 2k1 − k2 − k5 dt VH (t) Ve (t) VOH (t) − k8

Table 1 Diffusion coefficients [16].

1  1010

1. 2.

NH (t) NHO 2 (t) VHO 2 (t)

+ k7 VH (t)

− k13

NH3 O + (t) Ne (t) VH3 O + (t) Ve (t)

NH (t) NH2 O 2 (t) VH2 O 2 (t) + k16 VH (t)

NOH (t) NH2 (t) VOH (t) VH2 (t)

(16)

dNOH NOH (t) Ne (t) NOH (t) NH (t) = − k4 − k5 dt Ve (t) VH (t) − 2k 6 − k15

NOH (t) NOH (t) VOH (t)

NOH (t) NH2 O 2 (t) VH2 O 2 (t)

+ k13 VOH (t)

− k9 − k16

NH (t) NH2 O 2 (t) VH (t) VH2 O 2 (t)

NOH (t) NHO 2 (t) VHO 2 (t)

NOH (t) NH2 (t) VH2 (t) + k14 VOH (t)

Ne (t) NH2 O 2 (t) Ve (t) VH2 O 2 (t)

(17)

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J. Barilla et al. / Journal of Physics and Chemistry of Solids 78 (2015) 127–136

dNe Ne (t) NH (t) Ne (t) Ne (t) Ne (t) NOH (t) = − k2 − 2k 3 − k4 dt VH (t) Ve (t) VOH (t) − k7

dNH3 O + dt

Ne (t) NH3 O + (t)

− k14

VH3 O + (t)

NH3 O + (t) Ne (t)

= − k7 − k12

dt

Ne (t) NH2 O 2 (t) VH2 O 2 (t)

− k11

Ve (t)

dVH2

NH3 O + (t) NOH − (t) VOH − (t)

(18)

+ k14 VOH − (t)

dNH2 dt

= − k16

+ k17 VH3 O + (t)

NHO 2 (t) VHO 2 (t)

dVO 2− (19)

+ k2 VH2 (t) dNH2 O 2

VH (t)

dt dNHO 2 dt

= − k11

= − k8

VH (t)

NHO 2 (t) NHO 2 (t) VHO 2 (t)

VHO 2 (t) VHO 2 (t)

NH (t) NH (t) , VH (t)

(34)

NH (t) Ne (t) , Ve (t)

(35)

(22)

T(H + OH) = − k5

NHO 2 (t) VHO 2 (t)

(33)

All the above chemical reactions run in parallel according to Eqs. (16)–(24). Concurrently with chemical reactions the volume of any radical cluster increases according to Eqs. (25)–(33). This volume is being increased due to their diffusion into surroundings. To solve the above model, we have used Continuous Petri nets where the terms of the right-hand of ordinary differential equations have been implemented due to transitions expressed by the following equations:

T(H + e) = − k2

NHO 2 (t) NHO 2 (t)

NH (t) NOH (t) , VOH (t)

(36)

(23) T(H + HO 2 ) = − k 8

NHO 2 (t) NOH (t)

NH (t) NHO 2 (t) VHO 2 (t)

,

(37)

VOH (t)

− k17 NHO 2 (t) + k11VHO 2 (t)

VOH (t) VH2 O 2 (t)

⎛D 3 t ⎞ = 128 ⎜⎜ OH ⎟⎟ ⎝ π ⎠

⎛ D 3t ⎞ dVe = 128 ⎜⎜ e ⎟⎟ dt ⎝ π ⎠

dt

⎛D 3 t ⎞ HO 2 ⎟ = 128 ⎜⎜ ⎟ ⎝ π ⎠

T(H + H) = − 2k1

+ k10 VH2 O 2 (t)

(32)

N H3 O + (t) NO 2− (t) V H3 O + (t) VO 2− (t)

T(H + H2 O 2 ) = − k13

NH (t) NH2 O 2 (t) VH2 O 2 (t)

,

(38)

NOH (t) NH2 O 2 (t)

⎛D 3t ⎞ dVH = 128 ⎜⎜ H ⎟⎟ dt ⎝ π ⎠

dVH3 O +

(31)

⎛ D 3− t ⎞ O2 ⎟ = 128 ⎜⎜ ⎟ ⎝ π ⎠

NOH (t) NOH (t) VOH (t) VOH (t)

+ k17 VO 2− (t)

− k9

(21)

Ve (t)

VHO 2 (t) VH (t)

NHO 2 (t) NH (t)

⎛D 3 t ⎞ H2 O 2 ⎟ = 128 ⎜⎜ ⎟ ⎝ π ⎠

NH2 O 2 (t) Ne (t)

NHO 2 (t) NH (t)

VH3 O + (t)

+ k15 VHO 2 (t)

NH (t) NH (t) VH (t) VH (t)

+ k 6 VH2 O 2 (t)

NO 2− (t) NH3 O + (t)

− 2k10

dVOH dt

− k14

VOH (t)

+ k 8 VH2 O 2 (t)

dNO 2−

+ k1VH2 (t)

NH2 O 2 (t) NOH (t)

− k15

dt

(20)

Ne (t) NH (t) Ne (t) Ne (t) + k 3 VH2 (t) Ve (t) VH (t) Ve (t) Ve (t)

NH2 O 2 (t) NH (t)

= − k13

dt

Ve (t) VH2 O 2 (t)

dt

dVHO 2

Ne (t) NH2 O 2 (t)

VOH (t)

(30)

VO 2− (t)

Ne (t) Ne (t) Ne (t) NOH (t) + k 4 VOH − (t) Ve (t) Ve (t) Ve (t) VOH (t)

NH2 (t) NOH (t)

dVH2 O 2 dt

NH3 O + (t) NO 2− (t)

NOH − (t) NH3 O + (t) dNOH − Ne (t) NH (t) = − k12 + k2 VOH − (t) dt VH3 O + (t) Ve (t) VH (t) + 2k 3 VOH − (t)

⎛D 3 t ⎞ H2 ⎟ = 128 ⎜⎜ ⎟ π ⎠ ⎝

⎛ D 3 +t ⎞ H3 O ⎟ = 128 ⎜⎜ ⎟ ⎝ π ⎠

⎛ D 3 −t ⎞ dVOH − = 128 ⎜⎜ OH ⎟⎟ dt ⎝ π ⎠

(24)

(25)

(26)

(27)

T((HH3)O + + e) = k7 VH (t)

H) T((OH + H2 ) = k16 VH (t)

NH3 O + (t) Ne (t) VH3 O + (t) Ve (t) NOH (t) NH2 (t) VOH (t) VH2 (t)

,

,

(39)

(40)

T(OH + e) = − k 4

NOH (t) Ne (t) , Ve (t)

(41)

T(OH + H) = − k5

NOH (t) NH (t) , VH (t)

(42)

T(OH + OH) = − 2k6

(28) T(OH + HO 2 ) = − k 9

NOH (t) NOH (t) , VOH (t) NOH (t) NHO 2 (t)

(29) T(OH + H2 O 2 ) = − k15

VHO 2 (t)

(43) ,

NOH (t) NH2 O 2 (t) VH2 O 2 (t)

(44) ,

(45)

J. Barilla et al. / Journal of Physics and Chemistry of Solids 78 (2015) 127–136

T(OH + H2 ) = − k16

NOH (t) NH2 (t)

T((HOH+) H2 O 2 ) = k13 VOH (t)

) T((eOH + H2 O 2 ) = k14 VOH (t)

T(e + H) = − k2

,

VH2 (t)

(46)

NH (t) NH2 O 2 (t) VH (t) VH2 O 2 (t) Ne (t) NH2 O 2 (t) Ve (t) VH2 O 2 (t)

,

T(H2 O 2+ e) = − k14

Ne (t) NOH (t) , VOH (t)

(51)

Ne (t) NH3 O + (t) VH3 O + (t)

,

Ne (t) NH2 O 2 (t) VH2 O 2 (t)

NH3 O + (t) Ne (t) Ve (t)

(52)

,

NH3 O + (t) NO 2− (t)

T(OH −+ H3 O +) = − k12

VO 2− (t)

NOH − (t) NH3 O + (t) VH3 O + (t)

−) T((eOH + e)

Ne (t) Ne (t) = 2k 3 VOH − (t) , Ve (t) V(e) (t)

=

k 4 VOH − (t)

−

T((HH2+) H)

(56)

VOH (t)

,

O2 ) T((HO = k17 VO 2− (t) 2)

T(HO 2+ H) = − k 8

VHO 2 (t) VH (t)

(69)

,

(70)

NHO 2 (t) NHO 2 (t) VHO 2 (t) VHO 2 (t)

NO 2− (t) NH3 O + (t) VH3 O + (t) NHO 2 (t) VHO 2 (t)

,

,

(71)

(72)

,

VH (t)

(73) ,

NHO 2 (t) NOH (t)

T(HO 2+ HO 2 ) = − 2k10

(68)

NHO 2 (t) NH (t)

NHO 2 (t) NH (t)

T(HO 2+ OH) = − k 9

,

VOH (t)

(74) ,

NHO 2 (t) NHO 2 (t) VHO 2 (t)

(75) ,

(76)

T(HO 2 ) = − k17 NHO 2 (t),

,

Ve (t) VH2 O 2 (t)

NH (t) NH (t) = k1VH2 (t) , VH (t) VH (t)

T((eH+2 )e) = k 3 VH2 (t)

,

Ne (t) NH2 O 2 (t)

NH2 (t) NOH (t)

T((eH+2 )H) = k2 VH2 (t)

(55)

(77)

(57) 2) T((HHO − = k11VHO 2 (t) 3O+ + O ) 2

NH3 O + (t) NO 2− (t) VH3 O + (t) VO 2− (t)

,

(78)

(58) HO 2 ) T((OH + H2 O 2 ) = k15 VHO 2 (t)

NOH (t) NH2 O 2 (t) VOH (t) VH2 O 2 (t)

(59)

Ne (t) NOH (t) , Ve (t) V(OH) (t)

) − T((eOH + H2 O 2 ) = k14 VOH (t)

T(H2+ OH) = − k16

,

VOH − (t)

T(e + H)

−) T((eOH + OH)

(54)

NH3 O + (t) NOH − (t)

Ne (t) NH (t) = k2 VOH − (t) , Ve (t) V(H) (t)

(OH −)

(53)

(67)

NOH (t) NOH (t) , VOH (t) VOH (t)

H2 O 2 ) T((HO = k10 VH2 O 2 (t) 2 + HO 2 )

−

,

(66)

,

VOH (t)

H2 O 2 ) T((HO = k 8 VH2 O 2 (t) 2 + H)

T(O 2−+ H3 O +) = − k11

,

NH2 O 2 (t) NOH (t)

(49)

T(e + OH) = − k 4

T(H3 O ++ OH −) = − k12

Ve (t)

H2 O 2 ) T((OH + OH) = k6 VH2 O 2 (t)

(50)

T(H3 O ++ O 2− ) = − k11

NH2 O 2 (t) Ne (t)

(48)

Ne (t) Ne (t) , Ve (t)

T(H3 O ++ e) = − k7

VH (t)

(47)

T(e + e) = − 2k 3

T(e + H2 O 2 ) = − k14

NH2 O 2 (t) NH (t)

T(H2 O 2+ OH) = − k15

,

Ne (t) NH (t) , VH (t)

T(e + H3 O +) = − k7

T(H2 O 2+ H) = − k13

133

,

,

(79)

(60)

⎛D 3t ⎞ TVH = 128 ⎜⎜ H ⎟⎟ , ⎝ π ⎠

(80)

(61)

⎛D 3 t ⎞ TVOH = 128 ⎜⎜ OH ⎟⎟ , ⎝ π ⎠

(81)

⎛ D 3t ⎞ TVe = 128 ⎜⎜ e ⎟⎟ , ⎝ π ⎠

(82)

⎛ D 3 +t ⎞ H3 O ⎟, TV H3 O + = 128 ⎜⎜ ⎟ ⎝ π ⎠

(83)

⎛ D 3 −t ⎞ TVOH − = 128 ⎜⎜ OH ⎟⎟ , ⎝ π ⎠

(84)

(62)

(63)

Ne (t) NH (t) , Ve (t) VH (t)

(64)

Ne (t) Ne (t) , Ve (t) Ve (t)

(65)

134

J. Barilla et al. / Journal of Physics and Chemistry of Solids 78 (2015) 127–136

⎛D 3 t ⎞ H2 ⎟, TVH2 = 128 ⎜⎜ ⎟ π ⎠ ⎝

Table 3 Comparison of the calculated values with experimental results.

⎛D 3 t ⎞ H2 O 2 ⎟, TVH2 O 2 = 128 ⎜⎜ ⎟ ⎝ π ⎠

(85)

Substance

Initial yield (G0)

Experimental yield (G)

Petri nets (G)

(86)

1. 2. 3.

H• OH• e− aq

0.42 5.5 4.78

0.62 2.8 2.8

0.62 2.82 2.8

4.

H3O+ H2 H2 O2

4.78

2.8

2.8

0.15 0

0.47 0.73

0.44 0.73

5. 6.

⎛ D 3− t ⎞ O2 ⎟, TVO 2− = 128 ⎜⎜ ⎟ ⎝ π ⎠

(87)

⎛D 3 t ⎞ HO 2 ⎟. TVHO 2 = 128 ⎜⎜ ⎟ ⎝ π ⎠

(88)

Transitions (34)–(79) cause changes of species number and transitions (80)–(88) cause changes of corresponding cluster volumes. We can see graphical representation of preceding mathematical expressions in Fig. 2. The graphical model is very simple and clear, which allows us to solve very complex systems considered in radiobiology. The model simulation using Continuous Petri nets is very fast, which allows us to study the influence of all model parameters. The evolution of clusters characterized by initial conditions (volume V0 (t0 ) and numbers of corresponding species) is then determined by diffusion coefficients D H , DOH , De , D H3 O+ , DOH −, D H2 , D H2 O2, DO2−, D HO2 and rate constants k1 − k17 established on the basis of experimental data taken from the literature (see Tables 1 and 2). The proper initial conditions will be given by the initial radical numbers determined by transferred energy while their ratio established in the literature for a given kind of ionizing particles will be respected. The other initial condition is represented by cluster volume that should be further tested. The system of ordinary differential equations will be solved with the help of Continuous Petri nets by the tool Visual object net þ þ (see [20]). As the result of solution we shall obtain the time dependencies of radical numbers corresponding to given cluster energy values and their concentrations expressed in form of graphs (see Section 4). It can be very helpful in analyzing the influence of chemical stage to radiobiological mechanism.

4. Petri nets model and experimental data In this paper we shall derive first the time dependent numbers of radicals (H•, OH•, e−aq , H3 O+ and H2) starting from yield values G0 corresponding to cluster energy of 100 eV immediately at the end of physically chemical stage (in irradiating deoxygenated water by photons emitted by Co60 nuclei) having been presented, e.g. by Buxton [10] and LaVerne and Pimblott [17]:

NOH • = 5.5,

Ne = 4.78,

NH3 O + = 4.78,

H OH

NH2 = 0.15.

As to the initial cluster volume the diameter value 16 nm corresponding to the energy of 100 eV has been inserted. This value is in good agreement with our earlier fit [6] obtained for clusters forming DSBs in DNA molecules under similar conditions: cluster diameter of approximately 27 nm; and transmitted energy of 300 eV. We have already studied the effect of the chemical phase on the DSB formation on the basis of differential equations (introduced in Section 2) (see [3–7]). In these papers, however, the results have been gained under some substantial simplifications (e.g. only main chemical reactions have been considered). Now all possible radicals and chemical species will be taken into account. Some

6

e H3O

5 number of radicals

NH • = 0.42,

characteristics will be then derived with the help of Petri nets; the given results will be compared to experimentally established values of average numbers of individual radicals remaining in clusters in the time of their ends in the case of deoxygenated water medium when the Co60 radiation has been used. In our mathematical analysis we have started as already mentioned from the initial cluster values (i.e., G0 yield values of individual radicals and initial volume size) corresponding to a transmitted energy of 100 eV. The changing numbers of individual radicals in the dependence on cluster evolution time have been derived; diffusion coefficients and chemical reaction rates having been taken from Tables 1 and 2. The final values of radicals remaining in the end of cluster diffusion have been compared to experimental yield G values – see Table 3; good agreement has been obtained. The time changes of individual radical numbers in corresponding clusters during diffusion have been shown in Fig. 3. As to the chemical reactions the changing values of radical concentrations in individual clusters may be more important. These concentration values may also be important in evaluating the effect in DSB formation. Their time dependencies in 16 nm clusters at energy of 100 eV have been represented in Fig. 4. It is evident from the corresponding graphs that OH radicals and aqueous electrons possess the highest concentrations that decrease quickly especially for aqueous electrons. Main radiobiological effect is caused evidently by OH radicals as it is commonly assumed. We can see that the concentrations of all radicals go to zero at the cluster diameter of ca. 50 ns. The concentration values depend, of course, not only on radical numbers, but also on initial cluster size, that need not be estimated quite exactly. We also introduce, therefore, for comparison the time concentrations for initial cluster diameter of 20 nm at the same transferred energy value; see Fig. 5. In Figs. 6 and in 7 we have presented the concentration dependencies for cluster sizes of 16 nm and 20 nm at energy 60 eV, respectively.

H2 H2O2

4 3 2 1 0 0

10

20

30

40

50

time [ns]

Fig. 3. Number of radicals in dependence on time for cluster diameter 16 nm and energy 100 eV.

J. Barilla et al. / Journal of Physics and Chemistry of Solids 78 (2015) 127–136

3

H

4 3.5

concentration [mmol/dm ]

concentration [mmol/dm ]

4.5

135

H

3

OH

2.5

e HO2

2 1.5 1 0.5

OH

2.5

e

2

HO2

1.5 1 0.5

0

0

0

10

20

30

40

50

0

10

20

time [ns]

Fig. 4. Concentrations depending on time for cluster diameter 16 nm and energy 100 eV.

concentration [mmol/dm ]

2.5

50

Fig. 8. Concentrations depending on time for cluster diameter 27 nm and energy 300 eV.

OH

2

e

NH • = 1.26,

HO2

NOH • = 16.5,

1.5

Ne = 14.34,

1

NH3 O + = 14.34,

0.5

NH2 = 0.45, 0

10

20

30

40

50

time [ns]

Fig. 5. Concentrations depending on time for cluster diameter 20 nm and energy 100 eV. 3 H

concentration [mmol/dm ]

40

higher) the following initial average radical numbers have been obtained for efficient radical clusters:

H

0

2.5

OH

2

e HO2

1.5 1 0.5 0 0

10

20

30

40

50

time [ns]

Fig. 6. Concentrations depending on time for cluster diameter 16 nm and energy 60 eV. 1.4 concentration [mmol/dm ]

30 time [ns]

H

1.2

OH

1

e

0.8 HO2

0.6 0.4

which has corresponded approximately to the energy of 300 eV. The initial average size of clusters has been 27 nm which has corresponded to t0 = 13 ns . The decrease of radical numbers in corresponding clusters is represented in Fig. 8 their concentration goes practically to zero at t′ = 50 ns, which corresponds to the diameter of 50 nm. It is evident that the most DSBs are caused by radicals OH, while an individual cluster may be efficient practically in times shorter than 10 ns as at least two SSBs must be formed in close neighborhood in a corresponding DNA molecule; single radicals may form one SSB only. For comparison the time dependencies of radical concentrations for cluster diameter of 20 nm at the same energy 300 eV are also introduced in Fig. 9. If we compare these dependencies (see Figs. 8 and 9) it is evident that initial radical concentrations in 20 nm cluster are higher but decrease much more rapidly than in the case of the greater dimension. The efficiency of 27 nm clusters in DSB formation (at the same initial radical numbers) may be expected to be higher. It follows from our results that only clusters corresponding to energy of several hundreds eV may cause DSBs that may be responsible for actual DNA damage. The given results correspond (as it has been mentioned), to the case when the water has been deoxygenated. More detailed analysis involving different contents of oxygen in water is being prepared.

0.2

7 0

10

20

30

40

50

time [ns]

Fig. 7. Concentrations depending on time for cluster diameter 20 nm and energy 60 eV.

As already mentioned our main aim has concerned the influence of processes in chemical phase on the efficiency of DSB formation in DNA molecules. Our basic model has been applied to the data presented by Blok and Loman [9] concerning the formation of DSB in DNA molecules dissolved in water containing different oxygen amounts during irradiation by Co60 radiation [6]. Under the assumption of standard yield radical number ratios (shown

concentration [mmol/dm ]

0

H

6

OH

5 e

4 HO2

3 2 1 0 0

10

20

30

40

50

time [ns]

Fig. 9. Concentrations depending on time for cluster diameter 20 nm and energy 300 eV.

136

J. Barilla et al. / Journal of Physics and Chemistry of Solids 78 (2015) 127–136

5. Conclusion The presented mathematical model has enabled us to simulate the chemical stage of radiobiological mechanism and to obtain the time dependencies of the concentrations of radicals and other species in corresponding clusters. The Continuous Petri nets have made possible to study better the concurrent role of diffusion process and chemical reactions of individual radicals. The model may be easily extended to involve the influence of other species or radiomodifiers being present (at different concentrations) in water medium during irradiation. It may be helpful in studying the damage effect of individual radicals to DNA molecules and, consequently, also for the study of the radiobiological effect on various living cells (see e.g. [1,2]). In the presented paper we have assumed the spherical symmetry of the corresponding radical clusters. The given approach may also be easily generalized to assuming cylindrical symmetry at higher energy transfers. In such a case it will be sufficient to change mathematical expressions characterizing diffusion volume evolution. In this paper the previously proposed model based on Continuous Petri nets has been extended and verified by the experimental data. Further results are being prepared now in which the given model will be used in a more detailed analysis of the socalled oxygen problem (i.e., increased radiobiological effect under oxygen presence). The earlier analysis of the older experimental data presented by Blok and Loman (see [9]) published by us will be repeated and the corresponding evolutions of individual radicals in individual clusters will be demonstrated at different oxygen contents. We hope that the new results to our knowledge will contribute to the mechanism leading to increased formation of DSBs (or SSBs) in DNA molecules when oxygen is present.

Acknowledgments This work was supported by the Project LG130131 of Ministry of Education, Youth and Sports of the Czech Republic (Grant no. INGO II LG 13031).

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