A numerical study of synchronization in the process of

Cent. Eur. J. Phys. • 11(4) • 2013 • 440-447 DOI: 10.2478/s11534-013-0221-5

Central European Journal of Physics

A numerical study of synchronization in the process of biochemical substance exchange in a diffusively coupled ring of cells Research Article

Dragutin T. Mihailović1∗ , Igor Balaž2 , Ilija Arsenić1 1 Faculty of Agriculture, University of Novi Sad, Dositej Obradovic Sq. 8, 21000 Novi Sad, Serbia 2 Department of Physics, Faculty of Sciences, University of Novi Sad, Dositej Obradovic Sq. 5, 21000 Novi Sad, Serbia

Received 24 December 2012; accepted 31 March 2013

Abstract:

In this paper we numerically investigate a model of a diffusively coupled ring of cells. To model the dynamics of individual cells we propose a map with cell affinity, which is a generalization of the logistic map. First, the basic features of a one-cell system are studied in terms of the Lyapunov exponent, Kolmogorov complexity and Sample Entropy. Second, the notion of observational heterarchy, which is a perpetual negotiation process between different levels of the description of a phenomenon, is reviewed. After these preliminaries, we study how the active coupling induced by the consideration of the observational heterarchy modifies the synchronization property of the model with N=100 cells. It is shown numerically that the active coupling enhances synchronization of biochemical substance exchange in several different conditions of cell affinity.

PACS (2008): 87.16.dj, 05.45.Xt, 02.60.Cb Keywords:

active coupling • logistic equation • synchronization • multi-cell system © Versita sp. z o.o.

1.

Introduction

Understanding how local intra-cellular biochemical exchange processes and global features, like environment and system size, influence the robustness, adaptability and evolution of the collective behavior of multi-cell systems is one of the most challenging topics in the biology of complex systems today [1–5]. Information coupling and ∗

E-mail: [email protected]

the exchange of biophysical substances among the components of multi-cell systems are both driven by a range of intrinsic and extrinsic factors. Several authors have made significant contributions to the understanding of multicell system dynamics through studies of the stability of the synchronized state, which is required for robust functioning of the multi-cell system in the face of noise and perturbation [2–8]. However, these authors considered cells as completely uniform particles, without internal structure and without the ability to change their behavior. In actuality, it is well known that in natural conditions, bacterial cells spend

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Dragutin T. Mihailović, Igor Balaž, Ilija Arsenić

0.5

Xmax,p

0.4 0.3 0.2 0.1 0.0 0.0

Figure 2.

Figure 1.

Schematic diagram of a model of substance exchange in a system represented by a ring of coupled cells.

most of the time in the stationary phase which is (in contrast to the exponential phase) characterized by a decrease in growth rate, slowdown of all metabolic processes and increase in resistance to several stress conditions [9– 12]. Since these and many other processes in a cell are defined as diffusion-like, it is of great importance to see: (i) how these processes can be better represented in models, by introducing affinity in the diffusive coupling associated with biochemical substance exchange; and (ii) how intra-cellular dynamics are affected bythe perturbation of parameters that represent the influence of the environment, cell coupling and cell affinity. In considering these problems we have to include observational heterarchy, a challenging topic when dealing with complex systems. Essentially, observational heterarchy reveals that it is impossible to unambiguously determine to which subsystems an element belongs [13, 14]. It is based on the notion of an agent carrying the adjustment of measurement [15, 16]. Therefore, the dynamics of the complex system are articulated in terms of two kinds of dynamics, Intent and Extent dynamics, and the interaction between them, where Intent corresponds to an attribute of a given phenomenon and Extent corresponds to a collection of objects satisfying that phenomenon [13]. Gunji & Kamiura [13] comprehensively elaborated this concept by: (i) considering the essence of heterarchy, (ii) applying the idea of heterarchy to general phenomena, and (iii) proposing the idea of observational heterarchy. They underlined that the process of measurement and description cannot be separated from what is observed and measured, and that epistemology cannot be separated from ontology, resulting in a dynamical description and a dynamical ontology [17].

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Dependence of concentration xmax,p on the cell affinity p.

In this paper, we propose a map for diffusive coupling, which includes a formalization of the cell affinity. We then apply a form of observational heterarchy to a modeled multi-cell system represented by a ring of coupled cells (with the proposed coupling) and numerically show how the active coupling enhances the synchronization of biochemical substance exchange in several different conditions of cell affinity.

2. A map with cell affinity in the form of a generalized logistic map As noted above, the exchange of biochemical substance is considered a diffusion-like process. The dynamics of intracellular behavior are typically expressed as a logistic map Φ(x) = rx(1−x), where x is the concentration in a cell and r is a logistic parameter, 0.0 < r ≤ 4.0 [13, 18]. However, we use an alternative form of this map, which includes a parameter p that represents the cell affinity. By introducing this parameter we formalize an intrinsic property of the cell that includes (i) the affinity of genetic regulators towards arriving signals, which determine the intensity of the cellular response and (ii) the affinity for the uptake of signaling molecules (Fig. 1) [19]. From the logistic equation it follows that the level of intra-cell dynamics is most intensive for the concentration xmax,p=1 = 0.5, which comes from dΦ(x)/dx = r(1−2x) ≡ 0 and p = 1. This is a particular case of the condition dΦ(x, p)/dx = rpx p−1 (1 − 2x p ). It means that the level of concentration xmax,p when the intra-cellular dynamics is the most intensive depends on the cell affinity p i.e. xmax,p = 1/21/p . Calculating the R integral Φ(x, p) = rpx p−1 (1 − 2x p )dx we get Φ(x) = rx p (1 − x p ), where 0.0 < x ≤ 1.0 and 0.0 < p ≤ 1.0. We will call this ‘the map with the cell affinity’. Figure 2 shows that the intensity of the intra-cellular dynamics starts to grow when the cell affinity p is around 0.1. The cell dynamics are expressed here as a difference equa441

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Kolomogorov complexity, Kc

Lyapunov exponent, λ

A numerical study of synchronization in the process of biochemical substance exchange in a diffusively coupled ring of cells

1.0 0.5 0.0 -0.5 -1.0 -1.5 -2.0 -2.5 (a) -3.0 0.0 0.1

In accordance with [23], the Kolmogorov complexity Kc is determined based on the number of bits of the shortest computer program which is able to generate the sequence [22]. Complexity analysis of a time series {xi }, i = 1, 2, 3, . . . , N was done through the following steps:

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Sample entropy, SamEn

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• Step 2: The complexity counter c(n) is defined as the minimum number of distinct patterns contained in a given character sequence, and is a function of the length of the sequence n. The value of c(n) approaches an ultimate value b(n) as n approaches infinity, i.e. c(n) = O(b(n)), b(n) = n/log2 n. • Step 3: The normalized complexity measure Ck (n) = c(n)/b(n) = c(n)log2 n/n is calculated. Ck (n) represents the quantity of information contained in a time series, and is 0 for a periodic or regular time series and 1 for a random time series. For a non-linear time series, Ck (n) lies between 0 and 1.

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• Step 1: The time series is encoded by constructing a sequence S of the characters 0 and 1 written as {S(i)}, i = 1, 2, 3, . . . , n, according to the rule s(i) = (0, xi < x∗ ) and s(i) = (1, xi ≥ x∗ ) , where x∗ is a chosen threshold. Here, we used the mean P value of the time series, i.e. x∗ = N i=1 xi /N [24].

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The dependence on the cell affinity p of (a) the Lyapunov exponent, λ, (b) the Kolmogorov complexity, Kc , and (c) sample entropy (SampEn) of intra-cell dynamics, simulated by eq. (1).

tion, so we avoid the double approximation of (i) finding a differential equation to approximate an essentially discrete process (during the modeling stage) and then (ii) approximating that differential equation by a difference scheme for numerical computing purposes [13, 20, 21], i.e. p

p

Φ(xi,n ) = rxi,n (1 − xi,n ).

(1)

The dynamics of this map (eq. (1)) are governed by two parameters, p and r, which express cell affinity and the influence of the environment, respectively. We analyze this map using the Lyapunov exponent, λ, Kolmogorov complexity, Kc , and sample entropy (SampEn). We calculate λ as n−1 1X ln|rpx p−1 (1 − 2x p )|. n→∞ n i=0

λ = lim

(2)

We also compute the sample entropy SampEn (m, rs , N). This is a measure that quantifies regularity and complexity. It is an effective analytical tool for both deterministic chaotic and stochastic processes, and is particularly useful in the analysis of physiological, sonic, climatic and biological signals [25–27]. It is an estimate of the conditional probability that a subseries of length m, which provides a pointwise match within a tolerance rs to a time series including N datapoints, will also match the next point in that time series. The threshold factor or filter rs is an important parameter. In principle, with an infinite amount of data, it should approach zero. With finite amounts of data, or with measurement noise, the value of rs typically lies between 10 and 20 percent of the standard deviation of the time series [28]. To calculate the SampEn from a time series X = (x1 , x2 , . . . , xN ), one should follow these steps [26]: • Step 1: Form a set of vectors Xm1 , Xm2 , . . . , XmN−m+1 defined by Xi = (xi , xi+1 , . . . , xi+m−1 ) , i = 1, 2, . . . , N − m + 1. • Step 2: Calculate the distance between Xmi and j j Xm , d[Xmi , Xm ] as the maximum absolute difference between their respective scalar components: j d[Xmi , Xm ] = max |xi+k − xi−k |.

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k∈(0,m−1)

Dragutin T. Mihailović, Igor Balaž, Ilija Arsenić

• Step 3: For a given Xmi , count the number of j(1 ≤ j ≤ N − m + 1, j 6= i), denoted as Bi , such that d[Xim , Xjm ] ≤ rs . Then, for 1 ≤ i ≤ N − m, Bim (rs ) = Bi /(N − m + 1). • Step 4: Define Bm (rs ) n=M P m { Bi (rs )}/(N − m).

as:

Bm (rs )

=

i=1

• Step 5: Similarly, calculate Aim (rs ) as 1/(N −m+1) times the number of j(1 ≤ j ≤ N − m + 1, j 6= i), j i such that the distance between Xm+1 and Xm+1 is m m less than or equal to j. Set A (rs ) as A (rs ) = n=M P m { Ai (rs )}/(N − m). Thus, Bm (rs ) is the probai=1

bility that two sequences will match for m points, whereas Am (rs ) is the probability that two sequences will match m + 1 points. • Step 6: Finally, define SampEn(m, rs ) = lim {−ln[Am (rs )/Bm (rs )]},which is estimated by the n→∞

statistic SampEn(m, rs , N) = −ln[Am (rs )/Bm (rs )]. Figure 3 shows the variation of λ, Kc and SampEn against cell affinity p, with: (i) r randomly chosen in the interval (3.0 − 4.0), (ii) initial condition x0 = 0.25 and (iii) m = 5 and rs = 0.05. For each x, 104 iterations of the map (eq.1) are applied, and the first 103 steps are abandoned. Figures 3a - 3b show that for p < 0.2, there are (i) negative values of λ and (ii) Kc values close to zero, corresponding to stable conditions and low complexity. For higher values of p there is an increase in complexity and a frequent occurrence of regions with instability (λ > 0). Figures 3a and 3c show that SampEn follows the Lyapunov coefficient, i.e. takes values near zero when λ < 0. The above analysis indicates that, for the lower levels of affinity, the generalized logistic map with the cell affinity (eq. 1) can better simulate the biochemical substance exchange in the cell than the logistic equation can. This is particularly pronounced for p < 0.2.

3. Observational heterarchy and biochemical substance exchange between two cells Observational heterarchy consists of two sets of intralayer maps, called Intent and Extent perspectives, and inter-layer operations satisfying the following conditions: (1) the inter-layer operations inherit the mixture of intraand inter-layer operations and (2) there is a procedure by which the inter-layer operation can be regarded as an adjoint functor. If the inter-layer operation satisfies the

conditions (1) - (2), it is called a pre-functor [13]. According to [13], preserving the above composition occurs as follows: A pre-functor, hF i : Int → Ext is mapping a set X to a set hF iX , and a map Φ to a map f ∗ Φf, where f ∗ f(x) = x for all x in f(X ) with f(X ) : hF iX → X . In this sense we call f ∗ the pseudo-inverse of f. Because applying a pre-functor to a map is expressed as composition of maps, it satisfies the conditions (1) and (2). The approximation is defined by the assumption that f is a one-to-one map. If one accepts that the approximation f ∗ = f −1 holds, then a pre-functor can become a functor. Given two maps, Φ, Ψ : X → X , hF i(Φ)hF i(Ψ) = (f ∗ Φf)(f ∗ Ψf) = f ∗ Φ(ff ∗ )Ψf = = f ∗ Φ(ff −1 )Ψf = f ∗ ΦΨf = hF i(Φf).

(3)

However, there is inconsistency between Intent and Extent [13], illustrated, for example, in adaptive mutation in the Lactose operon [29–31]. Thus, in the phenomenon of the protein population, the Intent, given by an ordinary differential equation, ignores its differences, while Extent, consisting of individual proteins, focuses on differences. Their equivalence comes from the approximation alone, and otherwise cannot happen [13]. The time development of the of the intra-cellular dynamics xi,n , for two cells, is expressed as xi,n+1 = (1 − c)Φ(xi,n ) + f(Φ(xi,n )),

(4)

where: n is the time iteration, i, j = 1, 2, xi,n ∈ [0.0, 1.0], c is the coupling parameter (concentration of the substrate), f is the map representing the flow of the material from cell to cell, and Φ is one of maps in the pair (Ψ, Φ) whose composition is preserved by a pre-functor hF i [13]. Here, we apply the framework of an observational heterarchy to the two cell system. If Intent and Extent are denoted by Φ and Ψ, respectively, the time development of the concentration is expressed as xi,n+1 = (1 − c)Φ(xi,n ) + Ψ(xj,n ). For Ψ(X ) = f(Φ(x)), this expression is reduced to eq. (4). We perform our analysis following the procedure described in [13]. First, in this section we address the synchronization of the passive coupling for two cells given by eqs. (4) and (1), and then, in the next section, we will show that perturbation can modify the dynamics and enhance robust behavior, opened to emergence in a multi-cell system of active coupling. Synchronization is well-known collective phenomenon in various multi-component biological systems [2–4]. The exchange of information (coupling) among the components can be either global or local. This is also considered on 443

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Figure 4.

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Normalized frequency of synchronization,Fp (λ < 0) for system of two cells passively coupled (eqs. (2)-(4)) as a function of cell affinity p. An averaging was done over all values of coupling parameter c and logistic parameter r.

the cell level, for example, in mechanisms of (i) cell cycle synchronization [32] or (ii) intercellular biochemical substance exchange [33]. Here, we consider that chaotic systems are synchronized only when the largest Lyapunov exponent of the driven system is negative [32, 34]. It was calculated according to the approach in [34]. We studied the stability of the fixed point by linearizing a n ≥ 2 component coupled system, and obtain Zn+1 = ζn Zn where ζn is the Jacobian of this system evaluated in (0, 0, . . . , 0) and = (x1,n , x2,n , . . . , xN,n ). By iterating we obtain

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n Y

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(5)

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Frequency of synchronization, Fp

A numerical study of synchronization in the process of biochemical substance exchange in a diffusively coupled ring of cells

xi ,n+1 = (1 − c)Φ n ( xi ,n ) + Ψ n ( x j ,n )

Figure 5.

A pair of Intent and Extent maps in fluctuated active coupling expressed as eqs. (8a)-(8d). The left diagram represents the Intent map, Φ1 (xi,n ) with i = 0, 1, and the right diagram represents the Extent map, Ψ1 (xj,n ) with i = 0, 1. The Intent map is replaced by a discontinuous map f ∗ [13].

4. Simulations of active coupling in a multi-cell system In nature, microscopic biochemical substrates are perpetually influenced by stormy perturbations, and these perturbations affect not only the state but also the function of cells [36–38]. Therefore, we address the behavior of active coupling [13], and estimate whether the coupled map system described above can achieve synchronization under the influence of perturbations. The active coupling dynamics of the two-cell system used in the simulations are defined by the following equations: xi,n+1 = (1 − c)Φn (xi,n ) + Ψn (xj,n )

(8a)

(6)

Ψn+1 = fΦn f ∗ f

(8b)

Figure 4 depicts the normalized frequency of synchronization Fp (λ < 0) for system of two passively coupled cells (eqs. (1) and (4)), as a function of cell affinity p, averaged over all values of the coupling parameter c and logistic parameter r. The value of the normalized frequency of synchronization Fp is calculated as

Φn+1 = f ∗ Ψn+1

(8c)

and thus we get the Lyapunov exponent [35]  n 

Q

λ = lim ln

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Fp = P

Nn (λ < 0) P , Nn (λ < 0) + Np (λ > 0)

(7)

where Nn (λ < 0) and Np (λ > 0) are the numbers of negative and positive values of the Lyapunov exponent, respectively. These numbers were calculated for fixed values of p, with c and r changing in the intervals (0.0, 1.0) and (1.0, 4.0), respectively, with a step of 0.05. From this figure it is seen that for p > 0.2, Fp becomes lower, indicating a decrease of number of states which are synchronized.

Φ0 (xi,n ) = rxi,n (1 − xi,n ). p

p

(8d)

We note that the dynamical system defined by eqs. (1) and (4) is called the passive coupling, and that is the usual coupled map system. The active coupling can be approximated to passive coupling, where the approximation is defined by adjunction or the equivalence between Intent and Extent. Compared with passive coupling, the behavior of active coupling is much more complex [13]. In eqs. (8a) - (8d), because of a pseudo-inverse map, f ∗ , all calculations are defined to be approximations. In simulations, the Intent map was a discontinuous map, expressed by Φn+1 = f ∗ Ψn+1 . In order to see how perturbation enhances robust behavior in the framework of observational heterarchy in a multicell system represented by a ring of coupled cells (Fig.

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Dragutin T. Mihailović, Igor Balaž, Ilija Arsenić

XN1 = (A + B) • XN.

(9)

The elements in the matrices in eq. (9) are X N1i,n+1 = xi,n+1 , X Ni,n = xi,n , Ai,k = (1 − c)Φn (xi,n )δi,k

(10)

(6f) where p is randomly chosen; here the process of biochemical substance exchange in a multi-cell system exhibits a strong tendency towards synchronization, even though the logistic parameter r is in the chaotic region (r =4.0, 3.82 and 3.6). 1.0

Lyapunov exponent, λ

1), we consider the following model. In our approach, a cell moves locally in its environment without making long pathways. According to [33], the system of coupled difference equations for a set of N cells exchanging biochemical substance, can be written in the form of a matrix equation:

r =3.0

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k 6= 1, i = N

where i = 1, 2, . . . , N and δi,k is the Kronecker delta. Simulations with active coupling, defined by eqs. (8a)(8d), were performed with and without the perturbation given in Fig. 5. The results of the simulations are shown in Fig. 6. In this figure the Lyapunov exponent λ is plotted against the coupling parameter c for active coupling with perturbation (black line) compared to the passive coupling (red line), for different values of the cell affinity p and logistic parameter r. Simulations were performed with the ring of N = 100 cells. The Lyapunov exponent was calculated using eqs. (5) - (6) and the Jacobian of the system given by eqs. (10) - (11) representing the ring of N = 100 cells. In calculating λ, for each c from 0.0 to 1.0 with step 0.005, 104 iterations were applied for an initial state, and then the first 103 steps were abandoned. In order to see how the active coupling modifies the synchronization property of the model, we performed two kinds of simulations. Firstly, we used r = 4.0 and a fixed value of the cell affinity p (Figs.(6a) - (6c)); secondly, we used a randomly chosen p and a logistic parameter r with values of 4.0, 3.82 and 3.6, respectively (Figs.(6d) - (6f)). Figs.(6a) (6c) show that in the chaotic regime (r = 4.0), regardless of the value of p, the Lyapunov exponent is always positive (λ > 0) and therefore the process of biochemical substance exchange in a multi-cell system is always unsynchronized. However, the stormy perturbation disturbs this state (Figs.(6a) - (6c)). Although the logistic parameter is settledat r = 4.0 for chaotic behavior, the coupling parameter c tunes the interaction and leads to synchronization in some intervals, particularly for p = 1.0 and p = 0.5. This behavior is most pronounced in Figs.(6d) -

Lyapunov exponent, λ

Bi,k

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The Lyapunov exponent against number of cells N in the ring, for three values of r; c takes values in the interval (0,1) while p is randomly chosen for each c.

The dynamics of the coupled multi-cell system (eqs. (9) – (11)) are governed by four main parameters: the number of cells N (ring size), the coupling parameter c, the logistic parameter r and the cell affinity p. Here we present the collective dynamics of the multi-cell system by varying the number of cells from N= 1 to 100 for (i) c taking values in the interval (0,1) and (ii) p randomly chosen for each c. The simulations were performed for three values of r: 3, 3.7 and 4. The Lyapunov exponent was calculated as in previous experiments. Figures (7a) - (7c) depict the Lyapunov exponent, against number of cells N in the ring for three values of r. From this figure it is seen that, regardless of the number of cells, the process of biochemical substance exchange in a multi-cell system is much more synchronized for higher values of the logistic parameter 445

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A numerical study of synchronization in the process of biochemical substance exchange in a diffusively coupled ring of cells

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Diagram of Lyapunov exponent, λ, against coupling parameter c for the fluctuated active coupling defined by eqs. (8a)-(8d) - P (black line) compared to passive coupling - N (red line) for different values of cell affinity p and logistic parameter r. In (a)-(c) p takes the fixed values (1.0, 0.5, 0.2), while r = 4. In (d)-(f) p is randomly chosen, while r takes values 4.0, 3.82, and 3.6 respectively. Simulations were performed with the ring of N = 100 cells.

r. A similar simulation with the dynamics of the coupled multi-cell system was done in [5], but for just two parameters (N and c).

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(P) r = 4.0 p = randomly

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Lyapunov exponent, λ

Lyapunov exponent, λ

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Concluding remarks

In conclusion, this paper has presented the following: (1) we have proposed a map with cell affinity generalizing the logistic map to model the dynamics of individual cells; (2) basic features of this map have been studied in terms of the Lyapunov exponent, Kolmogorov complexity and Sample Entropy; (3) we have studied how the active coupling induced by considering observational heterarchy modifies the synchronization property of a model with N = 100 cells; (4) it has been shown that this active coupling enhances synchronization in several different conditions of cell affinity, i.e. although in simulations the logistic parameter is settled for chaotic behavior, the coupling pa-

rameter tunes the interaction and leads to synchronization; (5) we have considered the collective dynamics of the multi-cell system by varying the number of cells as well as the main parameters; (6) the results discussed above suggest that the proposed model of a diffusively coupled ring of cells can be used in simulating the process of substance exchange in a multi-cell system.

6.

Acknowledgments

This paper was realized as a part of the project Studying Climate Change and its Influence on the Environment: Impacts, Adaptation and Mitigation (43007), financed by the Ministry of Education and Science of the Republic of Serbia within the framework of integrated and interdisciplinary research for the period 2011 – 2014.

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Dragutin T. Mihailović, Igor Balaž, Ilija Arsenić

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