The use of The peTri neT meThod in The simulaTion modeling of miTochondrial swelling

Using photon correlation spectroscopy, which allows investigating changes in the hydrodynamic diameter of the particles in suspension, it was shown that ultrahigh concentrations of ca2+ (over 10 mm) induce swelling of isolated mitochondria. an increase in hydrodynamic diameter was caused by an increase of nonspecific mitochondrial membrane permeability to Ca ions, matrix Ca2+ overload, activation of atPand ca2+-sensitive k+-channels, as well as activation of cyclosporine-sensitive permeability transition pore. to formalize the experimental data and to assess conformity of experimental results with theoretical predictions we developed a simulation model using the hybrid functional Petri net method.


R
egulation of matrix volume is important for the mitochondria functioning and maintai ning of their integrity.The mitochondrial volume affects the respiration rate and the level of ATP production.There is a correlation between changes in the geometry of these organelles and such events as the generation of reactive oxygen species, the polarization of the inner mitochondrial mem brane, the ability to apoptosis.Mitochondrial swel ling is not only the final stage of their dysfunction, but also a key feature in the biochemical mecha nisms of cell damage.It leads to the straightening of the cristae, disruption of the outer membrane integ rity and release of apoptogenic factors such as cy tochrome c and apoptosisinducing factor, etc. into cytosol [14].
In vitro experiments on isolated mitochondria suspension standard approaches such as: inner mi tochondria membrane permeabilization by antibio tics, matrix Ca 2+ loading in the presence of Ca 2+ ionophore A-23187, induction of the non-specific membrane permeability by Ca ions and long chain saturated fatty acids, etc., to initiate swelling of the organelles are often used [5,6].In this work we pro pose another model approach for the initiation of increasing of non-specific permeability of the inner mitochondrial membrane and the matrix volume, namely the use of calcium ions in the ultrahigh (tens of mM) concentrations.In our view, this model for swelling of isolated mitochondria in vitro is less invasive and destructive.In particular, it does not apply the perforation of the inner membrane and a substantial chemical modification of the lipid bilayer.
To formalize and generalize the experimental data, to reduce the number of measurements and re place them with mathematical calculations, and also to assess conformity of experimental results with theoretical predictions we developed a simulation model, which connects the changes in mitochondria hydrodynamic diameter under effect of Ca 2+ and a conception of their structural and functional fea tures.As an approach for modeling of changes in mitochondria volumetric characteristics under the influence of Ca ions, ATP, cyclosporine, K + channel inhibitors, nitrocompounds, etc., we proposed to use hybrid functional Petri nets [7,8].
The advantages of the hybrid functional Petri nets as a modeling method are [9]: 1. Capability to structurally represent the states of the modeled system and the processes occurring in the system.
2. Quantitative modeling of three types of states and processes simultaneously: discrete, con tinuous and associative (forming).
3. Possibility to consider the activating, inhibi ting and catalytic effects by the means of a special type of bonds.
The process of swelling is commonly evaluated on the changes in optical density/light scattering of the mitochondrial suspension.The photon correla doi: http://dx.doi.org/10.15407/ubj88.04.066 tion spectroscopy technique is informative and ef ficient for analysis of the size of spherical-like parti cles in the solutions.This method is highly sensitive (the changes in volume from 0.1% can be registered and the particles with diameter from 0.001 to 20 µm can be analyzed); it requires a minor amount of ex perimental material, and also allows eliminating the artifacts associated with small size of subcel lular particles, which occur at application of other optical approaches.Photon correlation spectroscopy provides direct determination of the hydrodynamic diameter of isolated myometrium mitochondria (ef fective size) in experimental conditions [7,10,11].
Thus, the aim of the present work was to de velop a simulation model for swelling of isolated myometrium mitochondria in hypercalcium solution using Petri nets methodology and perform inhibito ry analysis of this process using photon correlation spectroscopy.

materials and methods
The preparation of isolated mitochondria was obtained from nonpregnant rat myometrium using differential centrifugation as described previously with minor modifications [12].After uterus removal and its purification from adipose and connective tissue the preparation was kept in 0.9% NaCl solu tion.Myometrium was cut with scissors into pieces (roughly 2x2 mm), which were put into the working solution with the following composition: 10 mM Hepes (pH 7.4), 250 mM sucrose, 1 mM EGTA at 4 °C.Tissue was homogenized with a homogenizer Polytron thrice for 20 s each time with cooling on ice for 1 min.The tissue:working solution ratio was 1 : 10.Homogenate was centrifuged at 1000 g for 15 min at 4 °C.Supernatant was centrifuged at 12 000 g for 15 min at 4 °C.The pellet was resus pended in the working solution and again centri fuged at 12 000 g for 15 min at 4 °C.The obtained isolated mitochondria were kept on ice during the experiment.
Protein concentration in the mitochondria fraction was determined by Bradford assay [13].Its avera ge value was 2 mg/ml and in the sample was 50 µg/ml.
To assess changes in the mitochondria volume we used photon correlation spectroscopy method, which allows determining their characteristic sizes (average hydrodynamic diameter).Particle volume in suspension was analyzed using correlation spec trometer ZetaSizer-3 (Malvern Instruments, UK) equipped with HeNe laser LGN111 (P = 25 mW, λ = 633 nm).Its operation principle is based on the analysis of the correlation characteristics of the fluc tuation intensity of dynamic light scattering while laser ray passes through a medium with mitochon dria.The measurements of the correlation function of the light scattering intensity fluctuations and in tegral scattering intensity enable to determine the translational diffusion coefficient for the dispersed particles in liquids, and to assess the particle size distribution using the StokesEinstein equation.Translational diffusion coefficient D is related to du ration of the correlation τ c with the ratio: Dq 2 = 1/τ с .
The wave vector of concentration fluctuations q is described by the expression: , where: n is the refractive index of the medium (liquid ); λ is the wavelength; θ is the scattering an gle.
Using the StokesEinstein equation that con nects a particle size, translational diffusion coeffi cient and viscosity of the liquid we can calculate a size (diameter) d(H) of spherical particles [7]: , where: k B is Boltzmann constant; t is absolute tem perature, °K; η is the shear viscosity of medium in which the particles are suspended; D is translational diffusion coefficient.
Recording and statistical processing of changes in the scattering intensity in the water (n = 1.33) mi tochondria suspension were performed for 10 min 10 times at +22 °C, at scattering angle 90°.The ob tained results were processed using the PCSSize mode v1.61 software.
For modeling we chose Cell Illustrator v.3 soft ware (Human Genome Center, University of Tokyo, Japan), the basic unit of which is a hybrid functional Petri nets [14].Petri net is a directed bipartite graph with two types of nodes (Table 1): places and tran sitions, which are connected by arcs, reflecting the structure of the net.Places usually characterize ob jects, elements, resources of the modeled system, transitions -the events that occur in the system and logical conditions of their implementation.

results and discussion
It was found that the values of mitochondria hydrodynamic diameter ranged from 500 to 700 nm with a mean of 550 ± 20 nm (n = 5, in experiments without cyclosporine) that corresponds well to the results of electron microscopy [11].As an example we represented a typical case of function of mito chondria distribution in suspension in the presence of cyclosporin A, a compound that reduces the pos sible variation of size deviations from average values by blocking permeability transition pore (PTP) (Fig. 1).In the left part of the distribution function there are albumin, other globular proteins and lipo protein complexes that are a minor part of the total number of objects.
We used Ca ions as agents that reliably trig ger organelle swel ling.The peculiarity of this re search was registration of the effects of ultrahigh nonphysio logical concentrations of this cation: 10 25 mM.For comparison, the local Ca 2+ concentra tion in the contact areas between the endoplasmic reticulum and mitochondria can reach more than 0.1 mM [15].However, we did not apply additional non-physiological agents, which artificially increase the nonspecific permeability of inner mitochondrial membrane.In the absence of Ca 2+ in the medium we did not observe statistically significant increase in the volume of isolated organelles during 10 min in both working sucrose solution (Fig. 2) and in the presence of physiological concentrations of K + .Swelling process induced by 20 mM Ca 2+ reached the steady state within 10 min.A substan tial increase in mitochondrial volume was observed only in a medium containing 1525 mM Ca 2+ (Fig. 2, 3) in the absence of MgATP 2 (obligatory condition).Increase in the hydrodynamic diameter appeared to be dependent on the Ca 2+ concentration: the latter changes from 0 to 25 mM followed by a significant increase in characteristic organelle sizes, and also the saturation on the cation concentration occurred.The barrier function of the inner mitochondrial membrane was so potent that even 10 mM Ca 2+ did not cause a notable increase in the hydrodynamic di ameter (Fig. 3).
Ruthenium red (10 µM) is a dye that reacts with biomembrane mucopolysaccharides and sialic acids [1618], and is considered as an inhibitor of Ca 2+ transport of mitochondrial inner membrane [15,19].It effectively inhibited the swelling in both mediums with sucrose and potassium chloride under action of 20 mM Ca 2+ (Fig. 4).
Obviously, the increasing of hydrodynamic dia meter is related not only to Ca 2+ influx into mito chondria through the Ca 2+ uniporter, since it is not only stimulated, but also effectively suppressed in the presence of MgATP 2 complex, which promotes the electrophoretic mitochondria Ca 2+ accumulation (graph data are not shown).It is known that ruthe nium red does not inhibit selectively enough Ca 2+ uniporter, but also affects other mitochondrial Ca 2+ transport system [15,19] and even Ca 2+ channels of endoplasmic reticulum [20].This feature is may be attributed to the non-specific modification of the subcellular membrane structures.Thus, we attribu ted increasing of mitochondria swelling upon the action of exogenous ultrahigh Ca concentrations to the increase in non-specific inner membrane perme ability to this cation followed by an increase in its concentration in the matrix.
A high concentration of ionized calcium in the matrix along with low level of ATP led to the opening of PTP and the osmotic imbalance be tween the matrix and the extramitochondrial me dium [3,21].In the presence of MgATP 2 complex (3 mM, equimolar amounts of Mg 2+ and ATP 4 ) and under the experimental conditions the swelling was not observed.The latter indicates that the mitochon drial ATP-sensitive K + channels, which are probably activated by Ca 2+ , were involved in the process of increasing of the matrix volume.It was also shown that Mg ions (13 mM) themselves did not affect the mitochondria hydrodynamic diameter and did not protect organelles from swelling.However, Ca 2+ induced increase in matrix volume was eliminated by 5 µM cyclosporine (graph data are not shown) in dicating a significant influence of cyclosporine sensi tive PTP on the studied process.It was assumed that the mechanism of the Ca 2+ dependent mitochondrial swelling includes both activation of ATPsensitive K + channels, and to a greater extent PTP involve ment in disturbance of the osmotic balance.
The idea of a possible role of K + permeability of inner mitochondrial membrane in the mechanisms for an increase in the organelles hydrodynamic dia meter in hypercalcium medium prompted us to study the influence of well-known blockers of K + channels on the swelling process.Due to the lack of sufficient concentration of K + (experiments with sucrose medium), a known nonselective inhibitor of K + channels 1 mM tetraethylammonium did not pre vent swelling of organelles (Fig. 4, a).In the presen ce of physiological K + concentration (132 mM), which is a prerequisite for the functional activity of mitochondrial K + channels, an increase in the hydro dynamic diameter under the effect of 20 mM Ca 2+ was more pronounced than in the sucrose medium (Fig. 4).This can be explained by the activation of K + transport through the channel structure into the matrix followed by mitochondria osmotic imbalance upon the action of ultrahigh concentrations of Ca 2+ , which is probably able to activate certain subtypes of K + -channels.Significant inhibition of the organelles swelling (Fig. 4, B) was observed when using non selective inhibitors of K + channels such as tetraethyl ammonium and 4aminopyridine (1 mM), blocker of Ca 2+ -dependent K + channels charybdotoxin (20 nM) and blocker of ATP-sensitive K + channels glibencla mide (20 µM).
The obtained results indicate a possible in volvement of various subtypes of K + channels in the mitochondria swelling under the action of exoge nous Ca 2+ in ultrahigh concentrations.Probably, the Ca 2+ overload along with inhibition of ATP gene ration lead to activation of mitochondria Ca 2+ and ATP-sensitive K + channels, and an increase in the concentration of K ions in the matrix, in turn, will contribute to the overall osmotic imbalance and or ganelles swelling.
Along with this, NO donor sodium nitro prusside (0.1 µM) effectively inhibited the Ca 2+ dependent swelling of myometrium mitochondria (graph data are not shown) that may be explained by the inhibitory effect of NO on PTP [22].Using the hybrid functional Petri nets we car ried out modeling imitating the influence of Ca 2+ (time and concentration patterns) on the value of the hydrodynamic diameter of myometrium mitochon dria (Fig. 5).
The model takes into account the components of mitochondria incubation medium (H 2 O, ATP, suc cinate), activators of mitochondria swelling (Ca, K ions) and the corresponding inhibitors (cyclosporine, sodium nitroprusside).In particular, we considered the following facts: (1) -an increase in the perme ability of the mitochondrial membrane to Ca ions under the influence of the hyper-calcium medium and the corresponding increase in the Ca ion con centration in the matrix; (2) -activation by Ca ions corresponding Ca 2+ -dependent K + channel subtypes; (3) -inhibition by ATP corresponding ATPsensi tive K + channel subtypes and PTP; (4) -the inner mitochondrial membrane depolarization upon in tensification of K + and Ca 2+ transport into matrix on the concentration gradient; (5) -opening PTP upon conditions of depolarization; (6) -blocking of PTP by cyclosporin A and sodium nitroprusside; (7)disorder of osmotic imbalance between the matrix and the extramitochondrial medium due to the PTP activation and followed by H 2 O transport into the matrix and increasing of organelle characteristic sizes; (8) -a role of succinate as a substrate acting at II complex level of electron transport chain.All of these processes are structurally indicated on the diagram (Fig. 5).
Through modeling we obtained mathematical equations, which formalize the process of mitochon dria swelling in the medium with ultrahigh concen trations of Ca 2+ .In particular, these equations can adequately describe the time characteristics of the process (Fig. 2).
According to Fig. 2 dynamics of the changes in the average values of mitochondria hydrodynamic diameter during swelling can be approximated by a polynomial of the third degree: D = At 3 -Bt 2 + Ct + D 0 , where: A = 2.476 nm/min 3 , B = 44.84nm/ min 2 , C = 303.6 nm/min -model coefficients; D 0 = 650 nm -average hydrodynamic diameter of mitochondria in control.
Since the swelling was caused by osmotic un balance between the matrix and the medium mostly due to the PTP activation, the permeability of PTP was the main model parameter.It is mathematically described as the time derivative of D: dD/dt = Et 2 -Ft + С, where: E = 7.428 nm/min 3 , F = 89.68nm/min 2 , C = 303.6 nm/min -model coefficients; t -time, min We also modeled the concentration (as Ca 2+ ) patterns of swelling based on the results presented in Fig. 3.The structural component of the model is similar to that shown in Fig. 5, but PTP permeability is described by the form: dD/dC m = HC m 3 + JC m 2 -KC m , where: H = 0.163 nm/mM 4 , J = 6.68 nm/mM 3 , K = 41.08 nm/mM 2 -model coefficients; C m -Ca 2+ concentration, mM.Our model enables to predict the changes in the organelle hydrodynamic diameter in time that signifi cantly optimizes the experimental procedures (time, consumption of reagents and laboratory ani mals), allows analyzing the dynamics of the process and comparing the results of modeling with actual observations under conditions of changes in the above parameters (composition of the incubation medium, the presence of activators/inhibitors).
Thus, in this work it has been demonstrated that ultrahigh concentrations of Ca 2+ (over 10 mM) induced mitochondrial swelling, which did not oc cur in the presence of MgATP 2 and cyclosporin A in the medium.Inhibitory analysis showed that this effect is caused by increa sing of non-specific mito chondrial membrane permeability to Ca ions, Ca 2+ overloaded matrix, activation of ATP and Ca 2+ sen sitive K + channels as well as PTP.
The use of Petri nets enables to structurally represent these processes, to consider activating and inhibitory action of medium components and to quantitatively model the mitochondria swelling in the real experimental conditions.acknowledgments We are deeply grateful to Member of NAS of Ukraine, Professor S. O. Kosterin for valuable ad vice during the discussion of the experimental re sults and the process of writing the manuscript.

T a b l e 1 .
The main structural elements of the hybrid functional Petri nets (see the text for the explanations)

Fig. 1 .Fig. 2 .
Fig. 1. the mitochondria size (volume) distribution in the presence of 5 µm cyclosporin as a stabilizing compound in sucrose medium Fig. 3. Mitochondria swelling induced by exogenous Ca 2+ at various concentrations in a sucrose medium; M ± m, * changes are statistically significant with respect to experiments with 0 mM Ca 2+ (P < 0.05)