# Experimental and Theoretical Analysis of Metal Complex Diffusion through Cell Monolayer

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Metal Complexes

_{6}](NO

_{3})

_{2}and [Co(1-allim)

_{6}](NO

_{3})

_{2}. The properties of these complexes were previously characterized with the use of crystallography and physicochemical analyses (infrated (IR), far-IR, Ultraviolet-Visible-near-IR (UV-vis-NIR) spectroscopy, magnetic moment, molar conductivity) [27].

#### 2.2. Cell Monolayer Construction

#### 2.3. Laser Interferometry

## 3. Results

#### 3.1. Diffusion of Metal Complexes Through a Eukaryotic Cell Monolayer

#### 3.1.1. Experiment

_{2}(3.08 × 10

^{−8}mol) was ~4.34 times greater than [Ni(1-allim)

_{6}](NO

_{3})

_{2}(7.09 × 10

^{−9}mol) after 60 min using 1-mL solutions at an initial concentration of 1 mM (p < 0.001). Cobalt(II) chloride was transported ~1.45 times better than the [Co(1-allim)

_{6}](NO

_{3})

_{2}complex (2.51 × 10

^{−8}mol and 1.73 × 10

^{−9}mol, respectively; p = 0.027). The diffusion coefficients of the tested compounds confirmed that both metal chlorides exhibited better diffusion properties than [Ni(1-allim)

_{6}](NO

_{3})

_{2}and [Co(1-allim)

_{6}](NO

_{3})

_{2}.

_{6}](NO

_{3})

_{2}complex when compared with the nontreated (control) cell monolayer. The same effect was observed for all tested metal chlorides and their complexes with 1-allylimidazole after 60 min.

#### 3.1.2. Theory

_{A}and σ

_{B}; σ

_{A}is a probability of a particle passing across the thin membrane over time τ from region A to B, and σ

_{B}is a similar probability for a particle moving in the opposite direction, and τ is the mean time needed for a particle to pass from point x = −d to x = 0 in the homogeneous diffusive system in which the membrane has been removed. Since the system consisting of a monolayer and nucleopore membrane is asymmetrical we accept that σ

_{A}≠ σ

_{B.}We assume that at the initial moment t = 0, all diffusing particles are in region A and the initial concentration of the particles is C

_{0}.

_{A}and σ

_{B}are the functions of time, which should be found from additional considerations.

_{A}and σ

_{B}do not change over time, we have, in a long time limit [30,31,32]:

_{A}= σ

_{B}).

_{A}and σ

_{B}by the time-dependent functions σ

_{A}―› σ

_{A}(t) and σ

_{B}―› σ

_{B}(t) in Equations (3) and (4). Using the trial and error method, we find that the following functions provide the best fit of the theoretical function to the empirical data in the long time limit:

_{0A}and σ

_{0B}are the initial permeability coefficients, and κ controls the time evolution of the coefficients; we assume that κ is the same for both functions. In Figure 3, we can see that the theoretical function N(t) coincides well with the empirical results for t > 25 min.

_{A}, σ

_{B}≤ 1 would not be met. Thus, we match the following function:

- for NiCl
_{2}, a = 4.25 × 10^{−9}mol/$\sqrt{\mathrm{min}}$ and b = 0, - for [Ni(1-allim)
_{6}](NO_{3})_{2}, a = 0.97 × 10^{−9}mol/$\sqrt{\mathrm{min}}$ and b = 0, - for CoCl
_{2}, a = 5.70 × 10^{−9}mol/$\sqrt{\mathrm{min}}$, b = 3.3 × 10^{−8}mol and κ = 0.14 1/min, - for [Co(1-allim)
_{6}](NO_{3})_{2}, a = 2.80 × 10^{−9}mol/$\sqrt{\mathrm{min}}$, b = 4.0 × 10^{−8}mol and κ = 0.14 1/min.

_{0B}/σ

_{0A}and does not depend on the values of the permeability parameters σ

_{A}and σ

_{B}for t > 0. This indicates that a molecule can pass through the surfaces of the barrier with certain probabilities, which may be different for both surfaces, but that inside the barrier the molecule diffuses ‘almost freely’. In other words, the obstacle makes it difficult for particles when they enter the barrier but not for their transport within the barrier. When b ≠ 0, the influence of the barrier on particle diffusion inside it is certainly larger than in the previous case.

_{0}= 1 mol/m

^{3}, S = 7 × 10

^{−5}m

^{2}and d = 1.5 × 10

^{−5}m.

#### Calculations for Nickel Compounds

_{0}= 1 mol/m

^{3}, the diffusion coefficient of NiCl

_{2}is D = 1.23 × 10

^{−9}m

^{2}/s [33]. Substituting the above values of parameters a,b extracted from the empirical data into Equation (6), we get η = 0.39, which provides σ

_{B}= 4.14σ

_{A}for nickel chloride. Assuming that η is the same for both nickel compounds, from Equation (6) we get D = 0.07 × 10

^{−9}m

^{2}/s for the nickel complex.

#### Calculations for Cobalt Compounds

_{0}= 1 mol/m

^{3}, the diffusion coefficient of CoCl

_{2}is D = 1.35 × 10

^{−9}m

^{2}/s. This coefficient was estimated from the results presented in [34]. Substituting the above values of parameters a, b extracted from the empirical data into Equations (6) and (7), we get η = 0.51, which provides σ

_{B}= 2.95σ

_{A}. Assuming that η is the same for the cobalt complex, from Equation (6) we get σ

_{A}= 2.04 × 10

^{−3}for cobalt chloride and σ

_{A}= 1.68 × 10

^{−3}for the cobalt complex. From Equation (6), we obtain D = 0.33 × 10

^{−9}m

^{2}/s for the cobalt complex.

_{A}, is larger for CoCl

_{2}than for [Co(1-allim)

_{6}](NO

_{3})

_{2}. Thus, smaller particles of cobalt chloride can more easily pass through the monolayer than larger cobalt complex particles.

## 4. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Yang, N.J.; Hinner, M.J. Getting Across the Cell Membrane: An Overview for Small Molecules, Peptides, and Proteins. Site-Specif. Protein Label. Methods Protoc.
**2015**, 1266, 29–53. [Google Scholar] [CrossRef] - Gravelle, S.; Joly, L.; Detcheverry, F.; Ybert, C.; Cottin-Bizonne, C.; Bocquet, L. Optimizing water permeability through the hourglass shape of aquaporins. Proc. Natl. Acad. Sci. USA
**2013**, 110, 16367–16372. [Google Scholar] [CrossRef] [Green Version] - Pagliara, S.; Dettmer, S.L.; Keyser, U.F. Channel-facilitated diffusion boosted by particle binding at the channel entrance. Phys. Rev. Lett.
**2014**, 113, 1–5. [Google Scholar] [CrossRef] [Green Version] - Nestorovich, E.M.; Danelon, C.; Winterhalter, M.; Bezrukov, S.M. Designed to penetrate: Time-resolved interaction of single antibiotic molecules with bacterial pores. Proc. Natl. Acad. Sci. USA
**2002**, 99, 9789–9794. [Google Scholar] [CrossRef] [Green Version] - Łapińska, U.; Glover, G.; Capilla-Lasheras, P.; Young, A.J.; Pagliara, S. Bacterial ageing in the absence of external stressors. Philos. Trans. R. Soc. B Biol. Sci.
**2019**, 374, 20180442. [Google Scholar] [CrossRef] - Thurber, G.M.; Yang, K.S.; Reiner, T.; Kohler, R.H.; Sorger, P.; Mitchison, T.; Weissleder, R. Single-cell and subcellular pharmacokinetic imaging allows insight into drug action in vivo. Nat. Commun.
**2013**, 4, 4–13. [Google Scholar] [CrossRef] [Green Version] - Pagliara, S.; Schwall, C.; Keyser, U.F. Optimizing diffusive transport through a synthetic membrane channel. Adv. Mater.
**2013**, 25, 844–849. [Google Scholar] [CrossRef] [Green Version] - Bezrukov, S.M.; Berezhkovskii, A.M.; Szabo, A. Diffusion model of solute dynamics in a membrane channel: Mapping onto the two-site model and optimizing the flux. J. Chem. Phys.
**2007**, 127, 1–9. [Google Scholar] [CrossRef] [Green Version] - Bauer, W.R.; Nadler, W. Molecular transport through channels and pores: Effects of in-channel interactions and blocking. Proc. Natl. Acad. Sci. USA
**2006**, 103, 11446–11451. [Google Scholar] [CrossRef] [Green Version] - Dagdug, L.; Vazquez, M.V.; Berezhkovskii, A.M.; Bezrukov, S.M. Unbiased diffusion in tubes with corrugated walls. J. Chem. Phys.
**2010**, 133, 127–130. [Google Scholar] [CrossRef] [Green Version] - Kolomeisky, A.B. Channel-facilitated molecular transport across membranes: Attraction, repulsion, and asymmetry. Phys. Rev. Lett.
**2007**, 98, 1–4. [Google Scholar] [CrossRef] [Green Version] - Dai, X.; Hou, C.; Xu, Z.; Yang, Y.; Zhu, G.; Chen, P.; Huang, Z. Entropic Effects in Polymer Nanocomposites. Entropy
**2019**, 21, 186. [Google Scholar] [CrossRef] [Green Version] - Xu, Z.; Gao, L.; Chen, P.; Yan, L.-T. Diffusive transport of nanoscale objects through cell membranes: A computational perspective. Soft Matter.
**2020**, 16, 3869–3881. [Google Scholar] [CrossRef] - Trejo-Solís, C.; Palencia, G.; Zuñiga, S.; Rodríguez-Ropon, A.; Osorio-Rico, L.; Torres Luvia, S.; Gracia-Mora, I.; Marquez-Rosado, L.; Sánchez, A.; Moreno-García, M.E.; et al. Cas Ilgly Induces Apoptosis in Glioma C6 Cells In Vitro and In Vivo through Caspase-Dependent and Caspase-Independent Mechanisms. Neoplasia
**2005**, 7, 563–574. [Google Scholar] [CrossRef] [Green Version] - Wąsik, S.; Arabski, M.; Dworecki, K.; Janoska, J.; Semaniak, J.; Szary, K.; Ślęzak, A. Laser interferometric analysis of glucose and sucrose diffusion in agarose gel. Gen. Physiol. Biophys.
**2014**, 33, 383–391. [Google Scholar] [CrossRef] [Green Version] - Kosztołowicz, T.; Dworecki, K.; Mrówczyński, S. How to measure subdiffusion parameters. Phys. Rev. Lett.
**2005**, 94, 6–9. [Google Scholar] [CrossRef] [Green Version] - Kosztołowicz, T.; Metzler, R.; Wa̧sik, S.; Arabski, M. Model of ciprofloxacin subdiffusion in Pseudomonas aeruginosa biofilm formed in artificial sputum medium. PLoS ONE
**2020**, 15. [Google Scholar] [CrossRef] - Kosztołowicz, T.; Wasik, S.; Lewandowska, K.D. How to determine a boundary condition for diffusion at a thin membrane from experimental data. Phys. Rev. E
**2017**, 96, 1–4. [Google Scholar] [CrossRef] [Green Version] - Arabski, M.; Wąsik, S.; Dworecki, K.; Kaca, W. Laser interferometric determination of ampicillin and colistin transfer through cellulose biomembrane in the presence of Proteus vulgaris O25 lipopolysaccharide. J. Membr. Sci.
**2007**, 299, 268–275. [Google Scholar] [CrossRef] - Gałczyńska, K.; Kurdziel, K.; Adamus-Białek, W.; Wąsik, S.; Szary, K.; Drabik, M.; Węgierek-Ciuk, A.; Lankoff, A.; Arabski, M. The effects of nickel(II) complexes with imidazole derivatives on pyocyanin and pyoverdine production by Pseudomonas aeruginosa strains isolated from cystic fibrosis. Acta Biochim. Pol.
**2015**, 62, 739–745. [Google Scholar] [CrossRef] [Green Version] - Arabski, M.; Lisowska, H.; Lankoff, A.; Davydova, V.N.; Drulis-Kawa, Z.; Augustyniak, D.; Yermak, I.M.; Molinaro, A.; Kaca, W. The properties of chitosan complexes with smooth and rough forms of lipopolysaccharides on CHO-K1 cells. Carbohydr. Polym.
**2013**, 97, 284–292. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Olszak, T.; Danis-Wlodarczyk, K.; Arabski, M.; Gula, G.; Maciejewska, B.; Wasik, S.; Lood, C.; Higgins, G.; Harvey, B.; Lavigne, R.; et al. Pseudomonas aeruginosa PA5oct jumbo phage impacts planktonic and biofilm population and reduces its host virulence. Viruses
**2019**, 11, 1089. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Danis-Wlodarczyk, K.; Vandenheuvel, D.; Jang, H.B.; Briers, Y.; Olszak, T.; Arabski, M.; Wasik, S.; Drabik, M.; Higgins, G.; Tyrrell, J.; et al. A proposed integrated approach for the preclinical evaluation of phage therapy in Pseudomonas infections. Sci. Rep.
**2016**, 6, 1–13. [Google Scholar] [CrossRef] [Green Version] - Danis-Wlodarczyk, K.; Olszak, T.; Arabski, M.; Wasik, S.; Majkowska-Skrobek, G.; Augustyniak, D.; Gula, G.; Briers, Y.; Jang, H.B.; Vandenheuvel, D.; et al. Characterization of the newly isolated lytic bacteriophages KTN6 and KT28 and their efficacy against Pseudomonas aeruginosa biofilm. PLoS ONE
**2015**, 10, e127603. [Google Scholar] [CrossRef] - Kosztołowicz, T.; Metzler, R. Diffusion of antibiotics through a biofilm in the presence of diffusion and absorption barriers. Phys. Rev. E
**2020**, 102, 1–11. [Google Scholar] [CrossRef] - Gałczyńska, K.; Ciepluch, K.; Madej, Ł.; Kurdziel, K.; Maciejewska, B.; Drulis-Kawa, Z.; Węgierek-Ciuk, A.; Lankoff, A.; Arabski, M. Selective cytotoxicity and antifungal properties of copper(II) and cobalt(II) complexes with imidazole-4-acetate anion or 1-allylimidazole. Sci. Rep.
**2019**, 9, 1–13. [Google Scholar] [CrossRef] - Kurdziel, K.; Glowiak, T. X-ray and spectroscopic characterisation of octahedral cobalt(II) and nickel(II) complexes with 1-allylimidazole in the solid state and electron-donor properties of the latter in aqueous solution. Polyhedron
**2000**, 19, 2183–2188. [Google Scholar] [CrossRef] - Schneider, C.A.; Rasband, W.S.; Eliceiri, K.W. NIH Image to ImageJ: 25 years of image analysis. Nat. Methods
**2012**, 9, 671–675. [Google Scholar] [CrossRef] - Arabski, M.; Wa̧sik, S.; Zych, M.; Łakomiec, W.; Kaca, W. Analysis of ciprofloxacin and gentamicin diffusion in Proteus mirabilis O18 biofilm by laser interferometry method. Acta Biochim. Pol.
**2013**, 60, 707–711. [Google Scholar] [CrossRef] - Kosztołowicz, T. Model of anomalous diffusion-absorption process in a system consisting of two different media separated by a thin membrane. Phys. Rev. E
**2019**, 99, 1–16. [Google Scholar] [CrossRef] [Green Version] - Kosztołowicz, T. Subdiffusion in a system consisting of two different media separated by a thin membrane. Int. J. Heat Mass Transf.
**2017**, 111, 1322–1333. [Google Scholar] [CrossRef] [Green Version] - Kosztołowicz, T. Random walk model of subdiffusion in a system with a thin membrane. Phys. Rev. E
**2015**, 91, 1–9. [Google Scholar] [CrossRef] [Green Version] - Ribeiro, A.C.F.; Gomes, J.C.S.; Barros, M.C.F.; Lobo, V.M.M.; Esteso, M.A. Diffusion coefficients of nickel chloride in aqueous solutions of lactose at T = 298.15 K and T = 310.15 K. J. Chem. Thermodyn.
**2011**, 43, 270–274. [Google Scholar] [CrossRef] - Ribeiro, A.C.F.; Lobo, V.M.M.; Natividade, J.J.S. Diffusion Coefficients in Aqueous Solutions of Cobalt Chloride at 298.15 K. J. Chem. Eng. Data
**2002**, 47, 539–541. [Google Scholar] [CrossRef] [Green Version] - Bergamo, A.; Sava, G. Ruthenium anticancer compounds: Myths and realities of the emerging metal-based drugs. Dalt. Trans.
**2011**, 40, 7817–7823. [Google Scholar] [CrossRef] - Kisova, A.; Zerzankova, L.; Habtemariam, A.; Sadler, P.J.; Brabec, V.; Kasparkova, J. Differences in the cellular response and signaling pathways between cisplatin and monodentate organometallic Ru(II) antitumor complexes containing a terphenyl ligand. Mol. Pharm.
**2011**, 8, 949–957. [Google Scholar] [CrossRef] [PubMed] - Chatterjee, S.; Kundu, S.; Bhattacharyya, A.; Hartinger, C.G.; Dyson, P.J. The ruthenium(II)-arene compound RAPTA-C induces apoptosis in EAC cells through mitochondrial and p53-JNK pathways. J. Biol. Inorg Chem.
**2008**, 13, 1149–1155. [Google Scholar] [CrossRef] - Chohan, Z.H.; Kausar, S. Synthesis, characterization and biological properties of tridentate NNO, NNS and NNN donor thiazole-derived furanyl, thiophenyl and pyrrolyl Schiff bases and their Co(II), Cu(II), Ni(II) and Zn(II) metal chelates. Met. Drugs
**2000**, 7, 17–22. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Rehman, S.; Ikram, M.; Rehman, S.; Shahnawaz, A. Synthesis, characterization and antimicrobial studies of Trnsition complexes of Imidazole derivatives. Bull. Chem. Soc. Ethiop.
**2010**, 24, 201–207. [Google Scholar] [CrossRef] [Green Version] - Arjmand, F.; Mohani, B.; Ahmad, S. Synthesis, antibacterial, antifungal activity and interaction of CT-DNA with a new benzimidazole derived Cu(II) complex. Eur. J. Med. Chem.
**2005**, 40, 1103–1110. [Google Scholar] [CrossRef] - Rodríguez-Argüelles, M.C.; López-Silva, E.C.; Sanmartín, J.; Pelagatti, P.; Zani, F. Copper complexes of imidazole-2-, pyrrole-2- and indol-3-carbaldehyde thiosemicarbazones: Inhibitory activity against fungi and bacteria. J. Inorg. Biochem.
**2005**, 99, 2231–2239. [Google Scholar] [CrossRef] [PubMed] - Shalini, K.; Sharma, P.; Kumar, N. Imidazole and its biological activities: A review. Chem. Sin
**2010**, 1, 36–47. [Google Scholar] - Congiu, C.; Cocco, M.T.; Onnis, V. Design, synthesis, and in vitro antitumor activity of new 1,4-diarylimidazole-2-ones and their 2-thione analogues. Bioorg. Med. Chem. Lett.
**2008**, 18, 989–993. [Google Scholar] [CrossRef] - Venkatesan, A.M.; Agarwal, A.; Abe, T.; Ushirogochi, H.; Ado, M.; Tsuyoshi, T.; Dos Santos, O.; Li, Z.; Francisco, G.; Lin, Y.I.; et al. 5,5,6-Fused tricycles bearing imidazole and pyrazole 6-methylidene penems as broad-spectrum inhibitors of β-lactamases. Bioorg. Med. Chem.
**2008**, 16, 1890–1902. [Google Scholar] [CrossRef] [PubMed] - Nakamura, T.; Kakinuma, H.; Umemiya, H.; Amada, H.; Miyata, N.; Taniguchi, K.; Bando, K.; Sato, M. Imidazole derivatives as new potent and selective 20-HETE synthase inhibitors. Bioorganic Med. Chem. Lett.
**2004**, 14, 333–336. [Google Scholar] [CrossRef] - Han, M.S.; Kim, D.H. Effect of zinc ion on the inhibition of carboxypeptidase A by imidazole-bearing substrate analogues. Bioorg. Med. Chem. Lett.
**2001**, 11, 1425–1427. [Google Scholar] [CrossRef] - Roman, G.; Riley, J.G.; Vlahakis, J.Z.; Kinobe, R.T.; Brien, J.F.; Nakatsu, K.; Szarek, W.A. Heme oxygenase inhibition by 2-oxy-substituted 1-(1H-imidazol-1-yl)-4-phenylbutanes: Effect of halogen substitution in the phenyl ring. Bioorg. Med. Chem.
**2007**, 15, 3225–3234. [Google Scholar] [CrossRef] [PubMed] - Nantermet, P.G.; Barrow, J.C.; Lindsley, S.R.; Young, M.; Mao, S.S.; Carroll, S.; Bailey, C.; Bosserman, M.; Colussi, D.; McMasters, D.R.; et al. Imidazole acetic acid TAFIa inhibitors: SAR studies centered around the basic P′1group. Bioorg. Med. Chem. Lett.
**2004**, 14, 2141–2145. [Google Scholar] [CrossRef] - Adams, J.L.; Boehm, J.C.; Gallagher, T.F.; Kassis, S.; Webb, E.F.; Hall, R.; Sorenson, M.; Garigipati, R.; Griswold, D.E.; Lee, J.C. Pyrimidinylimidazole inhibitors of p38: Cyclic N-1 imidazole substituents enhance p38 kinase inhibition and oral activity. Bioorg. Med. Chem. Lett.
**2001**, 11, 2867–2870. [Google Scholar] [CrossRef]

**Figure 2.**Scheme showing the laser interferometer and the experimental system used in this work. The barrier consisted of a PET membrane with a monolayer of CHO-K1 cells formed for 48 h at 37 °C with 5% CO

_{2}.

**Figure 3.**The amount of [Ni(1-allim)

_{6}](NO

_{3})

_{2}, [Co(1-allim)

_{6}](NO

_{3})

_{2}and metal chlorides transported through a monolayer of CHO-K1 cells formed on a PET membrane after 48 h at 37 °C with 5% CO

_{2}. Symbols represent empirical data, and solid lines represent theoretical functions.

**Figure 4.**Microscopy images of the CHO-K1 cell monolayer. (

**A**,

**C**) show optical microscopy images (100× magnification) of the cell monolayer stained by Giemsa before and after the diffusion measurement of the [Co(1-allim)

_{6}](NO

_{3})

_{2}complex (60 min), respectively. (

**B**,

**D**) show optical microscopy images (100× magnification) of the cell monolayer before and after the diffusion measurement of the [Co(1-allim)

_{6}](NO

_{3})

_{2}complex (60 min), respectively.

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**MDPI and ACS Style**

Gałczyńska, K.; Rachuna, J.; Ciepluch, K.; Kowalska, M.; Wąsik, S.; Kosztołowicz, T.; Lewandowska, K.D.; Semaniak, J.; Kurdziel, K.; Arabski, M.
Experimental and Theoretical Analysis of Metal Complex Diffusion through Cell Monolayer. *Entropy* **2021**, *23*, 360.
https://doi.org/10.3390/e23030360

**AMA Style**

Gałczyńska K, Rachuna J, Ciepluch K, Kowalska M, Wąsik S, Kosztołowicz T, Lewandowska KD, Semaniak J, Kurdziel K, Arabski M.
Experimental and Theoretical Analysis of Metal Complex Diffusion through Cell Monolayer. *Entropy*. 2021; 23(3):360.
https://doi.org/10.3390/e23030360

**Chicago/Turabian Style**

Gałczyńska, Katarzyna, Jarosław Rachuna, Karol Ciepluch, Magdalena Kowalska, Sławomir Wąsik, Tadeusz Kosztołowicz, Katarzyna D. Lewandowska, Jacek Semaniak, Krystyna Kurdziel, and Michał Arabski.
2021. "Experimental and Theoretical Analysis of Metal Complex Diffusion through Cell Monolayer" *Entropy* 23, no. 3: 360.
https://doi.org/10.3390/e23030360