Anna Y Gyurova
email: any_gyurova@abv.bg

Alexandar M Zhivkov

Influence of ethanol on the high frequency electric polarizability of E. coli

Keywords: Escherichia coli, Maxwell-Wagner polarizability, charge dependent electric polarizability, dielectric permittivity, electric conductivity, electro-optics

The interface electric polarizability of bacteria (charge dependent (ChD) and Maxwell-Wagner (MW) polarizabilities) gives information about their electric charge, determined by the structure and functional state. It is well known that the polarizability could be changed significantly by adding some substances to the suspension, and can be measured using an electro-optical (EO) method. There are some literature data, according to which the adding of ethanol decreases the electric polarizability of the cells. However the reason for the change in this parameter is not clear, as well as which component (ChD and/or MW) of polarizability has the main contribution. Generally the present work shows that the effect of ethanol is connected to the change of the internal (cytoplasm) MW polarizability and is mainly caused by increasing the cell membrane permeability. This results in an ionic flow through the membrane, which velocity and direction depends on the relative values of the inner (cytoplasm) and the outer medium ionic strength.

A C C E P T E D

M A N U S C R I P T

Introduction

The surface charge density and the electric polarizability of bacteria are important parameters giving information about the cells' structure and functional state [START_REF] Miroshnikov | Electro-physical analysis and separation of cells[END_REF]. These electric properties of bacteria, especially of E. coli, have been widely studied by some methods, the main of which are electrophoresis [START_REF] Miroshnikov | Electro-physical analysis and separation of cells[END_REF][START_REF] Jones | [END_REF], dielectric spectroscopy [3,[START_REF] Takashima | Electrical properties of biopolymers and Membranes[END_REF] and electro-optics [START_REF] Stoylov | Electro-optics of Colloids[END_REF][START_REF] Stoylov | Colloid electro-optics -Theory, Techniques and Application[END_REF][START_REF] Peikov | [END_REF]8]. The electrophoresis is the best developed method for determination the surface charge density, but it does not give any information about the electric charge inside bacteria, as the dielectric spectroscopy and electro-optics do. The last method has the advantage that it does not require high particle concentration contrary to the dielectric measurements.

The electro-optical methods are based on the orientation of colloid particles in applied electric field [START_REF] Stoylov | Electro-optics of Colloids[END_REF][START_REF] Stoylov | Colloid electro-optics -Theory, Techniques and Application[END_REF]. The orientation degree is proportional to the electric polarizability, which can be two main types: interface and volume. As a rule, the volume component is negligible in a suspension of particles with micron and submicron dimensions because of the big ratio between the total surface and the volume of all particles. Two basic kinds of interface polarizability are known: Maxwell-Wagner (MW) and surface charge dependent (ChD) polarizability.

The MW polarizability is defined as accumulation of electric charges at the interface between two mediums having different volume electric conductivity and dielectric permittivity under the action of electric field [START_REF] Stoylov | Electro-optics of Colloids[END_REF][START_REF] Stoylov | Colloid electro-optics -Theory, Techniques and Application[END_REF]. In the case of bacteria this polarization is due to accumulation of ions in the cytoplasm nearby the cytoplasm membrane, which is

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3 practically non-conductive. Therefore the MW component increases with the increase in the ionic strength inside the bacteria. MW polarizability does not depend on the particle size and its relaxation usually takes place in the MHz-range [START_REF] Stoylov | Colloid electro-optics -Theory, Techniques and Application[END_REF]9,10].

The ChD polarizability is defined as polarization of the particle electric double layer (EDL) [START_REF] Stoylov | Electro-optics of Colloids[END_REF], mainly of its diffuse part [11], and it is due to ions' migration parallel to the particle's surface. That is why its value depends on the bacteria long size and the EDL thickness determined by the surface charge density and the medium ionic strength. The ChD polarizability relaxation time depends on the square of the bacteria length and consequently may be observed at relatively low frequencies for cells with micron dimensions.

In the case of charged solid particles the ChD component plays the main role in their orientation, which is proved by the dependence of the electro-optical effect (EOE) on the pH and the ionic strength of the medium: the polarizability practically disappears at the isoelectric point and/or at high electrolyte concentration [START_REF] Stoylov | Electro-optics of Colloids[END_REF][START_REF] Stoylov | Colloid electro-optics -Theory, Techniques and Application[END_REF]. However in the case of bacteria the MW component should have the most significant contribution, because the cell volume is enclosed by a cytoplasm membrane and a cell wall with different electric parameters than the cytoplasm and the outer medium.

Usually in the electro-optical literature bacteria are treated like homogeneous solid particles [START_REF] Stoylov | Electro-optics of Colloids[END_REF][START_REF] Stoylov | Colloid electro-optics -Theory, Techniques and Application[END_REF][START_REF] Peikov | [END_REF][8] without considering their more complex structure: liquid cytoplasm inside and cell cover (cytoplasm membrane and cell wall) over it. That is why in most of the cases only the ChD polarizability is taken into account.

Because of the existence of two surfaces of the cell cover -external and internal ones, two EDL must be considered and two ChD polarizabilities -external (EChD) and internal (IChD) ones respectively must exist [12]. By analogy two MW components -external (EMW) and internal (IMW) ones should exhibit.

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bacteria, based on the MW type of polarization [START_REF] Miroshnikov | Electro-physical analysis and separation of cells[END_REF]. In that case the bacterium is considered like a homogeneous or hollow ellipsoid covered by one or more layers characterized by their specific electric conductivity and dielectric permittivity. The influence of the ionic atmosphere near the cell wall on the polarizability can be estimated by introduction of frequency-dependent values of the conductivity and the permittivity and characteristic frequencies for α and β dispersions [13]. This model shows a presence of few dispersion frequencies and different polarizability values along and across the bacterium [START_REF] Miroshnikov | Electro-physical analysis and separation of cells[END_REF]. It is possible to fit the theoretical frequency dependence with the experimental one by adjustment of the model parameters. The main disadvantage of this approach is that the EPh-model has too many parameters, which gives practically unlimited possibilities to fit the theoretical results to the experimental ones and consequently it is not clear if the parameters obtained in such a way are real.

We apply an experimental approach to distinguish the different types of surface polarization changing the factors influencing the ChD and the MW components. We use the electro-optical method electric turbidity to obtain the frequency dependence of the polarizability. Generally our approach reduces to investigation of the dependence of the electric polarizability on the bacteria size and the ionic concentration (inside or outside of the cells) at constant other experimental conditions.

In our previous article [START_REF] Zhivkov | High frequency electric polarizability of bacteria E. coli: Dependence on the medium ionic strength[END_REF] we have found out that the electric polarizability decreases with the increase in the outer ionic concentration (at constant one inside) in the frequency range from 20 kHz to 20 MHz. Our study has shown that the polarizability depends linearly on the DEL thickness, which is an indication for exhibition of the EChD polarizability even at these high frequencies, where the ChD component would disappear according to the dielectric and the electro-optical literature.

A C C E P T E D M A N U S C R I P T

In the present work we distinguish the IMW polarizability from the other components by changing the inner ionic strength. To this purpose we use ethanol, which increases the ion permeability of the cell membrane, possibly by making pores in it [START_REF] Angersbach | Electro-optical analysis of bacterial cells[END_REF].

Materials and Methods

Materials

Bacteria culture of E. coli K12 was cultivated in standard media LB and 1% glucose for 7.5 h at temperature 37 0 C and pH 7. The bacteria samples were taken from the medium just before the electro-optical measurements. After that they were washed with distilled water on a 0.8 μm milliporous filter and suspended in aqueous medium with electric conductivity of 5 μS/cm.

To obtain a suspension of fixed (dead) E. coli K12, bacteria were suspended in a fresh 3 % aqueous solution of formaldehyde after the described filtration. The bacteria were washed from the formaldehyde and suspended in distilled water just before the beginning of the electro-optical measurements.

The experiments connected to increasing in the cytoplasm ionic concentration (Fig. 6789) were carried out as follow: Two suspensions of fixed E. coli K12 have been incubated for 24 h in 150 mM KCl in absence and in presence of 5 % vol. ethanol respectively. After that bacteria were washed and suspended in distilled water just before the beginning of the electrooptical measurements.

In all the experiments the suspensions of alive or fixed bacteria were adjusted to an optical density 0.1 at 1 cm optical path.

Electro-optical theory

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When electric field is applied to the bacterial suspension, the optical density A is changed due to the particles' orientation. The value of the electro-optical effect (EOE) is determined as ΔA = A E -A, where the index E is for the applied electric field. In steady-state EOE (ΔA s ) the degree of orientation depends on the ratio between the orientation energy γЕ 2 and the energy of random motion kT of the particles with electric polarizability γ in electric field with intensity E:

(1)

ΔA s /A = G(F) ⋅ (γE 2 / 15kT) ,
where G(F) is an optical function, which depends on the size, form and refractive index of the cells.

The EOE decay after the switching off the electric field is defined by the rotational diffusion coefficient D, related respectively to the particle relaxation time τ =1/6D. The EOE decay of a mono-disperse system is mono-exponential for particles with axial symmetry:

(2)

ΔA t = ΔA s exp (-6Dt) = ΔA s exp (-t /τ) ,
where ΔA t and ΔA s are the values of EOE at the moment t and at the steady-state EOE respectively.

Electro-optical device

The main part of the electro-optical measurements was executed by EloTrace-1.0, developed in Biotronix GmbH (Germany). The device records the EOE as ΔA = ΔA // -ΔA ⊥ , where ΔA // and ΔA ⊥ are the values of the EOE at parallel and perpendicular orientation of two light beams towards the electric field. Because ΔA // and ΔA ⊥ have an opposite sign, the resulting EOE is a sum of both the components. EloTrace-1.0 takes down automatically the dependencies of the EOE on the frequency (20 kHz -20 MHz) and the intensity of the electric field (17 -110 V/cm) at wave length 670 nm and optical path 1 cm. It also calculates the average relaxation time of bacteria and their average size.

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7

The rest of the experiments were carried out by EloTrace-2.0 (Biotronix GmbH, Germany), which has the same electrical and optical characteristics, but makes completely automatically all the operations. The device takes a bacteria sample of 1 -2 ml from the incubator every 6 minutes, filtrates and washes it until obtaining a suspension with the definite optical density and electric conductivity mentioned above. After that EloTrace-2.0 takes down and treats the electro-optical data, so that the change in the electric polarizability and the average bacteria size with the growth time can be recorded.

All the frequency dependences of alive and fixed bacteria E. coli K12 are taken down at electric field intensity of 78 V/cm.

Results and discussion

In Fig. 1 are presented the frequency dependencies of the electric polarizability of fixed E. coli K12 at different ethanol concentrations. The value of this parameter decreases at all the frequencies of the experimental range and its relaxation frequency shifts to lower frequencies with the increase in the ethanol concentration. The electric polarizability decrease in the maximum of the dispersion curves is from 20 % to 50 % at ethanol concentration from 5 % vol. to 20 % vol. respectively.

The frequency dependencies of polarizability of alive bacteria E. coli K12 at different ethanol concentrations are shown in Fig. 2. A similar tendency in the electric polarizability behavior is observed as in the case of dead cells. We note that a comparison between the results for alive and fixed bacteria would not be correct in our case, because they come from different cultivations, and additionally this was not included in the aim of our study.

The decrease in the polarizability at adding of ethanol to E. coli suspension has been recorded by some researchers [START_REF] Angersbach | Electro-optical analysis of bacterial cells[END_REF][START_REF] Morris | [END_REF]. According to Ref. [START_REF] Morris | [END_REF] this fact is a proof for the interfacial origin of the dipole moments, however the study of the mechanism of polarizability

A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT 8
has not been an aim of the work. The other authors [START_REF] Angersbach | Electro-optical analysis of bacterial cells[END_REF] explain the polarizability decrease with the ethanol effect of damaging the cell wall by making pores in it.

In Fig. 3 and 4 are presented the dependencies of the electric polarizability of the maximum of frequency dependences of fixed (Fig. 1) and alive (Fig. 2) E. coli K12 on the ethanol concentration. The polarizability decreases linearly with the increase in the ethanol concentration.

The decrease in the electric polarizability (Fig. 1 and2) may be due to a change in the medium refractive index, the medium dielectric permittivity and the inner ionic strength.

The medium refractive index (n 0 ) increases from 1.333 to 1.347 at ethanol concentration increase up to 20 % vol. [START_REF] Ioffe | Refractometric methods in chemistry[END_REF], which leads to light scattering intense decrease and therefore to the optical density decrease. However the calculations using the theory of Mi [START_REF] Van De Hulst | Light Scattering by Small Particles[END_REF][START_REF] Kerker | The scattering of light and other electromagnetic radiations[END_REF] show that the increase of n 0 by 1 % results in an insignificant change of the turbidity [START_REF] Klenin | Characteristic functions of light scattering of dispersion systems[END_REF] at constant particles' dimensions. Furthermore, the change of n 0 does not influence the value of polarizability, which magnitude is determined by the relative value of the EOE:

ΔA s /A= A E /A-1, because the optical density at bacteria orientation in electric field A E and in the case of random orientation A are changed in the same way by n 0 . The ethanol effect also could be due to the change in the optical function G(F) (Eq. [START_REF] Miroshnikov | Electro-physical analysis and separation of cells[END_REF]), which depends on the relative particles' size B/λ, where the wave length in the medium λ is defined as λ= λ 0 /n 0 .

However this effect could not explain quantitatively the twice decrease in polarizability values observed in Fig. 1 and2.

Fig. 5 shows the dependence of the relative dielectric permittivity of the medium (ε) on the ethanol concentration. We can see that ε decreases linearly by 15 % (from 78 to 67) with the increase in the ethanol concentration from 0 to 20 % vol. (at 25 0 C) [START_REF] Dobos | Electrochemical data[END_REF]. The comparison between Fig. 5 and Fig. 3 and 4 results in the supposition that the decrease in the polarizability may be related to the decrease in ε of the medium at presence of ethanol.

A C C E P T E D M A N U S C R I P T

The calculations according to the EPh model for a homogenous ellipsoid [START_REF] Miroshnikov | Electro-physical analysis and separation of cells[END_REF] show that the electric polarizability decreases by 30 % in the frequency range from 10 3 to 10 7 Hz with the decrease in the medium ε from 80 to 60. This decrease is lower than that in our experiment, but the estimation presented above leads to the conclusion that the adding of ethanol influences significantly the value of the polarizability.

However the study of Ref. [START_REF] Angersbach | Electro-optical analysis of bacterial cells[END_REF] shows that there is a kinetics of the decrease in the polarizability with the time in presence of ethanol. The fact can not be explained by change in ε, because the decrease in this parameter of the medium happens at once at the moment of adding of ethanol, consequently the value of the polarizability should be decreased at once too, if it is only due to the change in ε. As it was mentioned, the authors suggest that the ethanol makes pores in the cell wall, through which an ionic flow appears. Because the ionic concentration inside the bacterium is higher than that in the outer medium in the case of Fig. 1 and 2, this flow should be directed out of the cell and would lead to decrease in the cytoplasm electrolyte concentration. The inner ionic concentration influences both the IMW and the IChD components.

According to the EO literature the components of the SChD polarizability should not exhibit at such high frequencies at such a large particle size [START_REF] Stoylov | Electro-optics of Colloids[END_REF]. However our previous study [START_REF] Zhivkov | High frequency electric polarizability of bacteria E. coli: Dependence on the medium ionic strength[END_REF] shows that the contribution of this component can be observed at the range from 20 kHz to 20 MHz, which means that the factor of the frequency is not enough for recognizing the polarizability mechanism. An additional criterion is the change in the ionic strength, which influences the different polarizability components in a different way, as it was described above. In our case the IChD polarizability should decrease strongly to disappearance because of the high electrolyte concentration of the cytoplasm. However the EChD component exists in the low ionic strength of the outer medium and maybe the ionic flow through the damaged

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membrane could increase the outer ionic strength. The following calculations help us to find if that supposition is possible:

The volume of a cell with average length 3 μm and diameter 1 μm is equal to 2.1 μm 3 .

The bacteria concentration in our experiment is 10 7 cells/cm 3 (at 0.1 OD), so the total volume of all the cells is 2.1⋅10 7 μm 3 in 1 cm 3 suspension. Even if all the ions flow out of the bacteria, the electrolyte concentration would be 5⋅10 4 -times lower than that in the cytoplasm (150 mM), therefore the outer medium concentration should become 3⋅10 -6 M. So, this estimation shows that the ionic flow through the ethanol damaged membrane is not able to change significantly the outer electrolyte concentration at the experimental conditions. Therefore in our case the magnitude of the EChD component remains almost constant and does not contribute to the decrease in the electric polarizability.

The EPh-model for homogeneous ellipsoid shows that the MW polarizability exhibits at electric conductivity values above 10 -2 S/m, which corresponds to 1 mM KCl. The outer electrolyte concentration in our experiment is less than 10 -5 M, therefore the EMW component does not exhibit. Anyway if it would exhibit, it would not be changed because the medium ionic strength is practically constant.

Generally we got to the conclusion that the decrease in the electric polarizability resulting from the ethanol effect of damaging the cell membrane is due to the decrease of the IMW component because of the progressively lower values of the cytoplasm electrolyte concentration.

Let us now turn back to Fig. 1 and2. As it was mentioned, the relaxation frequency of the polarizability shifts to lower frequencies with the increase in the ethanol concentration.

The decrease in the ionic strength leads to relaxation frequency shift to lower frequencies in the case of exhibition of ChD polarizability [START_REF] Petkanchin | Counterions dynamics as studied by Electric light scattering[END_REF], as well as of the MW one [START_REF] Dukhin | Dielectric phenomena and double electric layer in disperse systems and polyelectrolytes[END_REF]. However, as it was discussed above, The IChD and EMW components practically disappear at the So, our study shows that there are two main reasons for the decrease in the electric polarizability as an ethanol effect -the decrease in the medium dielectric permittivity and the ionic flow out of the cells. However both the factors lead to decrease in the polarizability and it is difficult to compare their contributions. Therefore, to separate them we had to change the experimental conditions in such a way that the two ethanol effects lead to opposite results.

To find out the relative contribution of the two ethanol effects, we made the following experiment. The bacteria E. coli K12 had been suspended in distilled water for a week in order to decrease strongly the inner ionic strength and then the ionic concentration of the suspension was increased up to 150 mM KCl (in ethanol absence). Another suspension was prepared and treated in the same way, but in presence of 5 % vol. ethanol. After the incubation under these conditions for 24 h both the suspensions were washed and suspended in aqueous medium and then investigated electro-optically (in ethanol absence).

The frequency dependencies of the electric polarizability of both the suspensions are presented in Fig. 6. The polarizability of the bacteria treated with KCl and ethanol is higher than that incubated only in KCl. In this experiment (in both the suspensions described) the outer electrolyte concentration was higher than the inner one, so the concentration gradient was directed from the medium to the cells. This ionic flow leads to increase in the cytoplasm ionic strength and therefore to higher value of the IMW component at the time of electrooptical measurement. On the other hand the presence of ethanol results in lower medium dielectric permittivity and decrease in the electric polarizability. Fig. 6 shows that the ethanol effect of increasing the inner ionic strength predominates over that of decreasing the medium dielectric permittivity.

A C C E P T E D M A N U S C R I P T Fig. 7 presents the kinetics of the change in the polarizability in the two suspensions with the time. We note that this experiment was carried out in aqueous medium (in ethanol absence). The cytoplasm electrolyte concentration of the bacteria treated by 5 % vol. ethanol is higher than that of the untreated bacteria. That is why the electric polarizability of the cells of the first suspension is higher in the beginning of the measurement. However the speed of the polarizability decreasing is different, because the ionic flow is faster in the case of ethanol-treated bacteria, whereas the ions' transport through the non-damaged membrane is slower.

Alternative explanation is that the decrease in the electric polarizability (Fig. 7) may be due to a possible decrease in the bacteria size with the time. That causes a decrease in the ChD component of polarizability, which still appears in the investigated frequency range [START_REF] Zhivkov | High frequency electric polarizability of bacteria E. coli: Dependence on the medium ionic strength[END_REF].

The relative change in the electric polarizability and the size of the cells with the time are compared in Fig. 8. Little changes in the bacteria size (which do not exceed 10 %) are observed. They result from a decrease in the osmotic pressure caused by the ionic transport out of the cells. However Fig. 8 shows that the discussed change in bacteria size corresponds to twice decrease in the value of the polarizability. Such an insignificant decrease can not cause a decrease in the polarizability like that observed in Fig. 7. So, the reason for the change in this parameter remains the decrease in the cytoplasm electrolyte concentration.

Summary

We found out that the bacteria electric polarizability decreases linearly with the increase in the ethanol concentration. The presented results can be explained by the influence of ethanol on the electric polarizability of bacteria in two ways -by decreasing the medium relative dielectric permittivity and increasing the membrane permeability (possibly by damaging the cell wall). The second factor causes gradual decreasing of the cytoplasm ionic

  experiment and the EChD one does not change. Consequently the relaxation frequency shift to lower values is only due to the change in the IMW polarizability.

Fig. 1 Fig. 2 Fig. 3

 123 Fig. 1 Frequency dependencies of the electric polarizability of fixed E. coli K12: ∎in

Fig. 4

 4 Fig. 4 Dependence of the electric polarizability of the maximum of the frequency

Fig. 5 Fig. 6

 56 Fig. 5 Dependence of the relative dielectric permittivity of the medium on the ethanol

Fig. 7 15 Fig. 8

 7158 Fig. 7 Kinetics of the change of the electric polarizability of the maximum of the frequency

Acknowledgment: We thank Dr. V. Bunin and Dr. A. Angersbach for the possibility, which they gave us to make the experiments in the laboratory of Biotronix GmbH, Germany, using the automatic electro-optical devices EloTrace 1.0 and EloTrace 2.0 developed in the same company.


Marko Gosak
email: marko.gosak@uni-mb.si

Cellular diversity promotes intercellular Ca 2+ wave propagation

Keywords: Ca 2+ waves, diversity-induced resonance, cellular variability, coupled cells

Calcium ions are an important second messenger in living cells. Calcium signals in form of waves serve as a means of intercellular communication and thus represent a vibrant subject for experimental and theoretical investigations. Here we study the role of cellular variability on the occurrence of Ca 2+ wave propagation in a net of diffusively coupled cells. Dynamics of individual cells is simulated by a mathematical model for Ca 2+ oscillations. Structural diversity of cells is introduced via variations of the bifurcation parameters, which signify cell sensitivity for external stimulation. Remarkably, for sufficient values of variability Ca 2+ waves emerge, which are mostly ordered for an intermediate variability strength. We analyze the spatial profile via the autocorrelation function, which confirms a resonance-like response due to the cellular variability.

Thus, the reported phenomenon is a novel observation of diversity-induced spatial coherence resonance in a tissue-like media.

A C C E P T E D M A N U S C R I P T A C C E P T E D M A N U S C R I P T Introduction

The impact of noise on nonlinear systems can cause a variety of interesting and counterintuitive phenomena [START_REF] Hänggi | Nonlinear Physics of Complex Systems[END_REF]. Perhaps the most famous is the phenomenon of stochastic resonance where an appropriate intensity of noise evokes the best correlation between a weak deterministic stimulus and the response of the nonlinear system [START_REF] Gammaitoni | Stochastic resonance[END_REF]. Even more remarkably, noise alone is also able to induce or enhance temporal order in the dynamics of a nonlinear system, whereby the phenomenon is to date most often addressed as coherence resonance [START_REF] Pikovsky | Coherence resonance in a noise-driven excitable system[END_REF]. Especially in biological systems, it is a rather firmly established fact that a certain degree of stochasticity is unavoidable [START_REF] Neiman | Noise in biophysical systems[END_REF]. Therefore, many studies have been devoted to the constructive role of noise in different biological processes. For instance, Douglass et al. [START_REF] Douglass | Noise enhancmet of information transfer in crayfish mechanoreceptors by stochastic resonance[END_REF] have provided experimental evidence that noise enhances the ability of information transfer in crayfish mechanoreceptor cells. Due to the indubitably presence of fluctuations in biological systems, several stochastic models has been developed for describing and studying different cellular processes. It has been shown that noise can induce stochastic Ca 2+ oscillations in cells operating near a Hopf bifurcation. These oscillations are mostly ordered for some intermediate values of noise [START_REF] Li | Internal noise stochastic resonance for intracellular calcium oscillations in a cell system[END_REF][START_REF] Zhu | A mesoscopic stochastic mechanism of cytosolic calcium oscillations[END_REF][START_REF] Chen | Selective effects of external noise on Ca 2+ signal in mesoscopic scale biochemical cell system[END_REF]. Similar studies have also been performed for circadian rhythms [START_REF] Yi | Enhancement of internal-noise coherence resonance by modulation of external noise in a circadian oscillator[END_REF] as well as genetic regulation [START_REF] Steuer | Constructive effects of fluctuations in genetic and biochemical regulatory systems[END_REF][START_REF] Wang | Engineered internal noise stochastic resonator in gene network: A model study[END_REF] and neuronal systems [START_REF] Lee | Coherence resonance in a Hodgkin-Huxley neuron[END_REF].

Following initial advances on systems without spatial degrees of freedom, the scope of interest shifted towards spatially extended media, where several noise-induced phenomena have been observed as well [START_REF] Sagués | Spatiotemporal order out of noise[END_REF]. The so-called spatiotemporal stochastic resonance has been reported in [START_REF] Jung | Spatiotemporal stochastic resonance in excitable media[END_REF], while spatial coherence resonance has been reported in several types of spatial extended systems, for example photosensitive Belousov-Zhabotinskii medium [START_REF] Perc | Noise-induced spatial periodicity in excitable chemical media[END_REF], in systems near pattern forming instabilities [START_REF] Carrillo | Spatial coherence resonance near pattern-forming instabilities[END_REF], and in excitable media [START_REF] Perc | Minimal model for spatial coherence resonance[END_REF][START_REF] Gosak | Spatial coherence resonance in excitable biochemical media induced by internal noise[END_REF]. Remarkably, the spatial coherence

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resonance phenomena has also been reported for diffusively coupled cells [START_REF] Perc | Periodic calcium waves in coupled cells induced by internal noise[END_REF], where it has been shown that internal stochasticity provokes a characteristic spatial frequency of calcium waves in the medium of coupled cells. Another interesting report was performed by Shuai and Jung [START_REF] Shuai | Sub-threshold Ca 2+ waves[END_REF],

who have studied the role spatial clustering of IP 3 receptors in a spatial extended model for intracellular calcium oscillations. Their results indicate that an inhomogeneous distribution of receptors, together with channel noise, warrant more coherent oscillations of Ca 2+ concentration.

The coherence resonance phenomenon has also been studied in more complex spatial extended systems. For instance, Sun et al. [START_REF] Sun | Spatial coherence resonance on diffusive and small-world networks of Hodgin-Huxley neurons[END_REF] have studied the spatial dynamics of Hodgin-Huxley neurons, whose interaction was governed by a small-world network. Remarkably, noise-induced spatial order has also been evidenced experimentally. Kádár et al. [START_REF] Kádár | Noise-supported traveling waves in sub-excitable media[END_REF] have shown that an appropriate intensity of noise sustain wave propagation in a photosensitive Belousov-Zhabotinsky reaction. Another interesting contribution has been reported by Jung et al. [START_REF] Jung | Noise-induced spiral waves in astrocyte syncytia show evidence of self-organized critically[END_REF], who have studied noise-induced calcium waves in astrocyte syncytia. Authors provided evidence that spatiotemporal sizes of single waves that are triggered by noise in a cultured network of glial cells, exhibit a power law distribution.

However, although the phenomena of stochastic and coherence resonance are most commonly associated with dynamic disorder or noise, only recently Tessone et al. [START_REF] Tessone | Diversity-induced resonance[END_REF] succeeded showing that static or quenched disorder can also provoke enhanced collective behavior of bistable and excitable oscillators. Their findings indicate, that many natural systems might profit from their intrinsic diversity. In this manner Chen et al. [START_REF] Chen | Structural-diversity-enhanced cellular ability to detect subthreshold extracellular signals[END_REF] engaged with analogous investigations in a cellular Ca 2+ signaling system. They have shown that the structural diversity of cells enhances the cellular ability to detect weak extracellular signals in a coupled hepatocytes systems. Specifically, the regularity of Ca 2+ oscillations exhibits a clear maximum with the variation of cellular

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diversity. Noteworthy, Glatt et al. [START_REF] Glatt | Variability-sustained pattern formation in subexcitable media[END_REF] have reported that variability is also able to induce pattern formation in a net of subexcitable units, where the most coherent patterns are achieved for intermediate values of parameter variability. In their further investigations [START_REF] Gassel | Doubly diversity-induced resonance[END_REF] interesting findings about the interplay of additive and multiplicative variability in a net of weakly forced bistable elements have been reported.

In the present paper we extend the scope of variability induced pattern formation to the medium of diffusively coupled cells, which local dynamics is driven by a mathematical model for Ca 2+

oscillations. We show that cellular variability in form of diverse cell-to-cell sensitivity to external stimulation induces Ca 2+ wave propagation, if the cells are operating near the Hopf bifurcation.

Remarkably, the most ordered waves are observed for intermediate values of cellular diversity.

Mathematical model

We use a simple mathematical model for Ca 2+ dynamics determined by theoretical framework of Goldbeter et al. [START_REF] Goldbeter | Minimal model for signal-induced Ca 2+ oscillations and for their frequency encoding through protein phosphorylation[END_REF] as a building block for our spatially extended system. The model considers changes of free Ca 2+ concentration in the cytosol ( z ) and in the intracellular calcium store ( y ).

Individual cells are arranged on a

L L × ( ] , 1 [ , L j i ∈
) square lattice and are diffusively coupled through the cytosol, which is modeled by an additional flux

j i z D , 2 ∇
. So, the spatial extension of the model has the following form:

j i j i j i j i j i z D z k y k v v v v dt dz , 2 , , f 3 2 1 , 0 , ∇ + - + + - + = β , ( 1
)
j i j i y k v v dt dy , f 3 2 , - - = , (2)
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where

n j i n n j i z K z V v , 2 , M2 2 + = , (3)
p j i p p j i m j i m m j i z K z y K y V v , A , , R , M3 3 + + = . ( 4
)
The Laplacian is incorporated into the numerical scheme via a five-point finite difference formula, as described by Barkley [29], using periodic boundary conditions. Parameter values, used in our calculations are quoted in the caption of Figure 1. Here the model equations are presented only briefly. For details, we refer to the original work of Goldbeter et al. [START_REF] Goldbeter | Minimal model for signal-induced Ca 2+ oscillations and for their frequency encoding through protein phosphorylation[END_REF]. We would only like to expose the term 1 , v j i β in Eq. ( 1), which denotes the IP 3 -modulated influx from the IP 3 sensitive store. In particular, the parameter j i , β measures the saturation of IP 3 receptors and is selected as the control parameter, which defines the calcium dynamics in the cell.

In order to introduce the cellular structural diversity, we assume that values of the control parameter j i, β are Gaussian distributed among cells and satisfy 0

β β = : ) 1 ( , 0 , j i j i σξ β β + = . (5)
In Eq. ( 5) σ defines the measure for structural diversity and j i,

ξ are uncorrelated random
Gaussian numbers with zero mean and unit variance. Clearly, there is a disparity in the cellular structure for 0 > σ , which in physiological meaning indicates diverse cell-to-cell sensitivity for external stimulation.

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Cellular diversity induced Ca 2+ waves

We set 39 . 0 0 = β in order to achieve that for 0 = σ all of the cells are just below the Hopf bifurcation, occurring at 397 . 0 0 ≈ β . In this case, all cells stay in a stable state and the whole lattice remains quiescent. On the other hand, by introducing the cellular variability, the control parameter j i , β differs from cell to cell and, as a result, some of the cells are placed into the oscillatory regime. In order to visualize the resulting spatial behavior, we show in Fig. 1 characteristic spatial profiles of j i z , for three different values of σ . For a small cellular diversity, the cells remain inactive. However, if σ is increased, ordered Ca 2+ waves are noticed that are propagating through the medium. As σ increases further, the coherence of waves becomes evidently destroyed.

Figure 1

The basis for the observed phenomenon is the proximity to the Hopf bifurcation point. As already mentioned above, for a non-zero structural diversity, some cells are put into the oscillatory regime. But, if σ is low, the number of oscillatory cells is too small to be capable of activating the other cells and inducing a coherent spatial dynamics. For intermediate values of σ , the fraction of the oscillatory cells in the medium becomes adequate and the steady-state cells are being triggered by small clusters of oscillatory cells, which induce and govern the spatial dynamics of ordered Ca 2+ waves. However, when σ increases further, the cells become more and more dispersed around the bifurcation point. Consequently, some of the cells are located too far from the bifurcation point and cannot be activated by their neighbors. In addition, several cells are situated in different regions of the oscillatory regime, which occasions different inherent frequencies and amplitudes of oscillations of individual cells. For that reasons larger values of σ

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obliterate the coherence of the waves. We can thus infer that an optimal cellular diversity should exist for the most coherent intracellular Ca 2+ wave propagation.

To quantify the spatial dynamics of Ca 2+ waves for different levels of cellular variability, we calculate the normalized autocorrelation function of the spatial domain j i z , :

2 , , , ) , (

j i j i j i z z z g η ξ η ξ + + = , (6) where )
( ~, , , j i j i j i z z z - = , j i z
, is the spatial average of the whole lattice at a particular time, and and η are representing spatial lag variables measured as a change in position from the starting i and j values. Figure 2 shows the results for the same parameters as employed in Fig. 1. As anticipated, for small values of σ , no considerable spatial correlation is noticed. On the other hand, for

, there is an uncontestable spatial correlation. For somewhat larger values of σ , the spatial correlation decreases, which is indicated by a greater decay of the autocorrelation.

Note, that the function ) , ( η ξ g shown in Fig. 2, represent the average over several temporal realizations.

Figure 2

Similar as by the characterization of temporal order with the classical temporal autocorrelation function, we calculate the spatial correlation length as:

∑∑ = = = 0 0 0 0 2 ) , ( ξ η λ i j j i g , (7)
in order to determine the optimal cell variability. Namely, the larger the value of λ , the larger the spatial order in the lattice. In the calculations, λ is obtained by averaging over 100 realizations.

Figure 3 features results that confirm a resonance-like behavior with increasing σ , since a clear maximum of the resonance curve can be noticed at 11 . 0 ≈ σ . In addition, we examine how the value λ depends on the distance from the Hopf bifurcation. We can observe, that the overall coherence of the spatial dynamics decreases as the distance from the bifurcation increases, and moreover, the peak of the resonance curves moves towards larger values of σ . Similar findings have been reported for the coherence resonance phenomenon, where the maximal temporal or spatial [START_REF] Perc | Periodic calcium waves in coupled cells induced by internal noise[END_REF] order that can be induced by noise was also find decreasing with the increasing distance from the Hopf bifurcation.

Discussion

Signals encoded in Ca 2+ waves regulate multiple cellular functions from cell fertilization to its death [START_REF] Berridge | Calcium -a life and death signal[END_REF] and are thus of key importance for normal functioning of living organism. Therefore, investigations of spatio-temporal calcium dynamics represent a vibrant topic in the scientific community. An interesting contribution has been reported by Tordjmann et al. [START_REF] Tordjmann | Coordinated intercellular calcium waves induced by noradrenaline in rat hepatocytes: dual control by gap junction permeability and agonist[END_REF][START_REF] Tordjmann | Receptor-oriented intercellular calcium waves evoked by vasopressin in rat hepatocytes[END_REF], who studied the coordination of intercellular Ca 2+ waves in rat hepatocytes. In their experimental investigations they hypothesize that an assembly of hepatocytes communicating via gap junctions might be regarded as an excitable medium. Furthermore, they discovered that heterogeneity of cell-to-cell sensitivity for the hormone plays a vital role in the coordination of Ca 2+ waves. Their study also signifies that agonist-induced intercellular waves are initiated by an IP 3 increase in the most sensitive cell, which provides a pacemaker-like mechanism for the regulation of intercellular communication in the liver. These findings were theoretically corroborated by

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Dupont et al. [START_REF] Dupont | Mechanism of receptor-oriented intercellular calcium wave propagation in hepatocytes[END_REF], who examined a model for Ca 2+ wave propagation, whereby several hepatocytes were differing in sensitivity to the agonist and thus in the intrinsic frequency of calcium oscillations. They revealed that wave propagation occurs in the direction of the gradient of hormonal sensitivity, which has also been confirmed experimentally.

in the present study we show that a random heterogeneity in cell-to-cell sensitivity for the external stimulation can provoke Ca 2+ waves in the medium of diffusively coupled cells. The coherence of the waves exhibits a resonance-like behavior with the variation of structural variability. The phenomenon is a novel observation of diversity-induced spatial coherence resonance in the medium of coupled cells. Remarkably, in the last few years the role of spatial heterogeneities on intercellular dynamics has been emphasized in investigations of cardiac tissues. Bub et al. [START_REF] Bub | Spiral wave generation in Heterogeneous excitable media[END_REF] have provided an explanation for the experimental observations of disturbances of calcium waves in chick embryonic heart cells as the coupling is reduced. On the basis of a simple model they found out that heterogeneity in the spatial arrangement plays the crucial role in wave organization. Recently, the importance of heterogeneities in cardiac tissue has also been emphasized by Woo et al. [START_REF] Woo | Spiral wave drift and complexoscillatory spiral waves caused by heterogeneities in two-dimensional in vitro cardiac tissues[END_REF], who have shown that inherent inhomogeneities in the cell culture influence the cardiac wave dynamics. Our results suggest that cellular variability might play an important role in intercellular calcium signaling or at least may influence other noise-induced effects that are inherent in biological systems. Therefore, when studying intra-and intercellular dynamics and organization, not only dynamical noise, but also cell heterogeneity should be considered.

Figure 3

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Figure 2

The average spatial correlation functions of j i z , obtained for the same parameters as in Fig. 1.

Figure 3

The dependence of the spatial correlation length on cellular variability for different values of 0 β .