# Fluctuation of Information Entropy Measures in Cell Image

^{*}

## Abstract

**:**

## 1. Introduction

_{i}stands for the frequency of pixels whose intensity values are within the range ($i$,$i+di$) and $di$ is an intensity difference that depends on the resolution of the camera or the possible number of gray values in the dynamic range of the pixels. Their formulation could be used to investigate the complexity or textural entropy of various mediums, including living ones. Changing the microscope or camera settings (for example increasing camera’s possible number of gray values, exposure time or microscope light intensity) is expected to influence $E$ values so that maintaining fixed microscope and camera settings is essential for comparing different media.

## 2. Materials and Methods

#### 2.1. Materials

#### 2.1.1. Cells

_{2}at 37 °C. Before use, exponentially growing cells were obtained, washed and suspended in PBS at a concentration of 1.5–2.0 × 10

^{6}cells/mL.

^{6}cells/mL. In all instances, more than 70% of the cells were defined as T-lymphocytes (CD3 positive). Viability, determined by PI staining, was always higher than 90%.

#### 2.1.2. Petri Dish-Based LCA (Live Cell Array) and Related Operations

^{6}cells/mL) in PBS or HEPES buffer, were loaded on top of the array and cells were allowed to settle by gravity for 5–10 min. Then, 500 µL of buffer was added gently to the medium exchange region around the array. The imaging dish was mounted on the microscope stage and measured.

#### 2.1.3. Induction of Cell Death and PI Staining

_{2}) and maintained for 12 h, then washed again and loaded on the array.

#### 2.2. Measurement System

#### 2.2.1. Microscope

#### 2.2.2. Image/Data Acquisition

#### 2.2.3. Fluctuation Based Measures

#### 2.3. Data Analysis and Statistics

## 3. Results and Discussion

#### 3.1. GLIE Fluctuation and Thermodynamic Entropy—A Theoretical Aspect

- Intracellular particle random motion and diffusivity in a time scale of 1–2 s is mostly not a result of thermal agitation but of random active mechanical fluctuations of the cytoskeleton and related intracellular content while in a thermodynamically non-equilibrium viscoelastic medium.
- The resulting diffusion might be characterized as “thermic-like”, with an approximately linear correlation between the square of translocations and time lags.
- Therefore, normal distribution describes the PDF of these intracellular particle translocations and the related energy levels as well.
- The above normal distribution PDF of total energy or translocations represents the thermodynamic generalized entropy of this particle system and hence correlates between diffusivity and thermodynamic generalized entropy (Equation (3)).

#### 3.2. Noise Due to Measurement System Fluctuations

#### 3.3. The Immediate Applicative Aspect of GLIE Fluctuation-Based Measures

## 4. Conclusions

## Supplementary Materials

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Kitano, H. Systems Biology: A Brief Overview. Science
**2002**, 295, 1662–1664. [Google Scholar] [CrossRef] [PubMed] - Tarabichi, M.; Antoniou, A.; Saiselet, M.; Pita, J.M.; Andry, G.; Dumont, J.E.; Detours, V.; Maenhaut, C. Systems biology of cancer: Entropy, disorder, and selection-driven evolution to independence, invasion and “swarm intelligence”. Cancer Metastasis Rev.
**2013**, 32, 403–421. [Google Scholar] [CrossRef] [PubMed] - Banjeri, C.R.; Miranda-Saavedra, D.; Severini, S.; Windschwendter, M.; Enver, T.; Zhou, J.X.; Teschendorff, A.E. Cellular network entropy as the energy potential in Waddington’s differentiation landscape. Sci. Rep.
**2013**, 3, 3039. [Google Scholar] [CrossRef] - Davies, P.C.; Rieperb, E.; Tuszynski, J.A. Self-organization and entropy reduction in a living cell. Biosystems
**2013**, 111, 1–10. [Google Scholar] [CrossRef] [PubMed] - Kullback, S.; Leibler, R.A. On information and sufficiency. Ann. Math. Stat.
**1951**, 22, 79–86. [Google Scholar] [CrossRef] - Guo, M.; Bao, E.L.; Wagner, M.; Whitsett, J.A.; Xu, Y. SLICE: Determining cell differentiation and lineage based on single cell entropy. Nucleic Acids Res.
**2017**, 45, e54. [Google Scholar] [CrossRef] [PubMed] - Cheng, F.; Liu, C.; Shen, B.; Zhao, Z. Investigating cellular network heterogeneity and modularity in cancer: A network entropy and unbalanced motif approach. BMC Syst. Biol.
**2016**, 10, 65. [Google Scholar] [CrossRef] [PubMed] - Maire, T.; Youk, H. Molecular-level tuning of cellular autonomy controls the collective behaviors of cell populations. Cell Syst.
**2015**, 1, 349–360. [Google Scholar] [CrossRef] [PubMed] - Vilar, J.M.G. Entropy of Leukemia on Multidimensional Morphological and Molecular Landscapes. Phys. Rev. X
**2014**, 4, 021038. [Google Scholar] [CrossRef] - Crofts, A.R. Life, information, entropy, and time: Vehicles for semantic inheritance. Complexity
**2007**, 13, 14–50. [Google Scholar] [CrossRef] [PubMed] - Luo, L. Entropy production in a cell and reversal of entropy flow as an anticancer therapy. Front. Phys. China
**2009**, 4, 122–136. [Google Scholar] [CrossRef] - Haralick, R.M.; Shanmugam, K.; Dinstein, I. Textural features for image classification. IEEE Trans. Syst. Man Cybern.
**1973**, SMC-3, 610–621. [Google Scholar] [CrossRef] - Pantic, I.; Pantic, S.; Paunovic, J. Aging increases nuclear chromatin entropy of erythroid precursor cells or cells in mice spleen hematopoietic tissue. Microsc. Microanal.
**2012**, 18, 1054–1059. [Google Scholar] [CrossRef] [PubMed] - Pantic, I.; Pantic, S.; Basta-Jovanovic, G. Gray level co-occurrence matrix texture analysis of germinal center light zone lymphocyte nuclei: Physiology viewpoint with focus on apoptosis. Microsc. Microanal.
**2012**, 18, 470–475. [Google Scholar] [CrossRef] [PubMed] - Pantic, I.; Pantic, S. Germinal center texture entropy as possible indicator of humoral immune response: Immunophysiology viewpoint. Mol. Imaging Biol.
**2012**, 14, 534–540. [Google Scholar] [CrossRef] [PubMed] - Pantic, I.; Pantic, S.; Paunovic, J.; Perovic, M. Nuclear entropy, angular second moment, variance and texture correlation of thymus cortical and medullar lymphocytes: Grey level co-occurrence matrix analysis. Anais Acad. Bras. Cienc.
**2013**, 85, 1063–1072. [Google Scholar] [CrossRef] [PubMed] - Gonzalez, R.C.; Woods, R.E. Digital Image Processing; Prentice Hall: Upper Saddle River, NJ, USA, 2008; ISBN-13: 978-0131687288. [Google Scholar]
- Wiedemann, P.; Guez, J.S.; Wiegemann, H.B.; Egner, F.; Quintana, J.C.; Asanza-Maldonado, D.; Filipaki, M.; Wilkesman, J.; Schwiebert, C.; Cassar, J.P.; et al. In situ microscopic cytometry enables noninvasive viability assessment of animal cells by measuring entropy states. Biotechnol. Bioeng.
**2011**, 108, 2884–2893. [Google Scholar] [CrossRef] [PubMed] - Schrödinger, E. What is Life? The Physical Aspect of the Living Cell; Cambridge University Press: Cambridge, UK, 1967. [Google Scholar]
- Wohl, I.; Zurgil, N.; Hakuk, Y.; Sobolev, M.; Galmidi, M.; Deutsch, M. In situ label-free static cytometry by monitoring spatiotemporal fluctuations of image gray values. J. Biomed. Opt.
**2015**, 20, 105013. [Google Scholar] [CrossRef] [PubMed] - Wohl, I.; Zurgil, N.; Hakuk, Y.; Sobolev, M.; Deutsch, M. In Situ Evaluation of Physiological Activity and Mitochondrial Dysfunction via Novo Label-Free Measures Based on Fluctuation of Image Gray Values. J. Anal. Bioanal. Tech.
**2016**, 7, 2. [Google Scholar] [CrossRef] - Paszek, P.; Jackson, D.A.; White, M.R.H. Oscillatory control of signalling molecules. Curr. Opin. Genet. Dev.
**2010**, 20, 670–676. [Google Scholar] [CrossRef] [PubMed] - Oancea, E.; Meyer, T. Protein kinase C as a molecular machine for decoding calcium and diacylglycerol signals. Cell
**1998**, 95, 307–318. [Google Scholar] [CrossRef] - Codazzi, F.; Teruel, M.N.; Meyer, T. Control of astrocyte Ca
^{2+}oscillations and waves by oscillating translocation and activation of protein kinase C. Curr. Biol.**2001**, 11, 1089–1097. [Google Scholar] [CrossRef] - Giri, L.; Patel, A.K.; Karunarathne, W.K.; Kalyanaraman, V.; Venkatesh, K.V.; Gautam, N. A G-protein subunit translocation embedded network motif underlies GPCR regulation of calcium osscillations. Biophys. J.
**2014**, 107, 242–254. [Google Scholar] [CrossRef] [PubMed] - Hatano, T.; Sasa, S. Steady State Thermodynamics of Langevin Systems. Phys. Rev. Lett.
**2001**, 86, 3463–3466. [Google Scholar] [CrossRef] [PubMed] - Saraste, A.; Pulkki, K. Morphologic and biochemical hallmarks of apoptosis. Cardiovasc. Res.
**2000**, 45, 528–537. [Google Scholar] [CrossRef] - Hakumäki, J.M.; Poptani, H.; Puumalainen, A.M.; Loimas, S.; Paljärvi, L.A.; Ylä-Herttuala, S.; Kauppinen, R.A. Quantitative 1H nuclear magnetic resonance diffusion spectroscopy of BT4C rat glioma during thymidine kinase-mediated gene therapy in vivo: Identification of apoptotic response. Cancer Res.
**1998**, 58, 3791–3799. [Google Scholar] [PubMed] - Montero, S.; Martin, R.R.; Guerra, A.; Casanella, O.; Cocho, G.; Nieto-Villar, J.M. Cancer Glycolysis I. Entropy Production and Sensitivity Analysis in Stationary State. J. Adenocarcinoma
**2016**, 1, 8. [Google Scholar] - Molnar, J.; Thornton, B.S.; Gábor, P. Thermodynamics and Information Physics Offer New Opportunities in Cancer Therapy. Curr. Cancer Ther. Rev.
**2014**, 10, 234–245. [Google Scholar] [CrossRef] - Guo, M.; Ehrlicher, A.J.; Jensen, M.H.; Renz, M.; Moore, J.R.; Goldman, R.D.; Lippincott-Schwartz, J.; Mackintosh, F.C.; Weitz, D.A. Probing the stochastic, motor-driven properties of the cytoplasm using force spectrum microscopy. Cell
**2014**, 158, 822–832. [Google Scholar] [CrossRef] [PubMed] - Sunray, M.; Zurgil, N.; Shafran, Y.; Deutsch, M. Determination of individual cell Michaelis-Menten constants. Cytometry
**2002**, 47, 8–16. [Google Scholar] [CrossRef] [PubMed] - Ravid-Hermesh, O.; Zurgil, N.; Shafran, Y.; Sobolev, M.; Galmidi, M.; Badihi, Y.; Israel, L.L.; Lellouche, J.P.; Lellouche, E.; Michaeli, S.; et al. Real-time quantification of protein expression and translocation at individual cell resolution using imaging-dish-based live cell array. Anal. Bioanal. Chem.
**2014**, 406, 7085–7101. [Google Scholar] [CrossRef] [PubMed] - MacKintosh, F.C.; Levine, A.J. Nonequilibrium mechanics and dynamics of motor-activated gels. Phys. Rev. Lett.
**2008**, 100, 018104. [Google Scholar] [CrossRef] [PubMed] - Mizuno, D.; Tardin, C.; Schmidt, C.F.; MacKintosh, F.C. Nonequilibrium mechanics of active cytoskeletal networks. Science
**2007**, 315, 370–373. [Google Scholar] [CrossRef] [PubMed] - Brangwynne, C.P.; Koenderink, G.H.; MacKintosh, F.C.; Weitz, D.A. Intracellular transport by active diffusion. Trends Cell Biol.
**2009**, 19, 423–427. [Google Scholar] [CrossRef] [PubMed] - Cartwright, J.H.E.; Piro, O.; Tuval, I. Fluid dynamics in developmental biology: Moving fluids that shape ontogeny. HFSP J.
**2009**, 3, 77–93. [Google Scholar] [CrossRef] [PubMed] - Shabaniverki, S.; Juárez, J.J. Characterizing gelatin hydrogel viscoelasticity with diffusing colloidal probe microscopy. J. Colloid Interface Sci.
**2017**, 497, 73–82. [Google Scholar] [CrossRef] [PubMed] - Saks, V.; Beraud, N.; Wallimann, T. Metabolic Compartmentation—A System Level Property of Muscle Cells. Int. J. Mol. Sci.
**2008**, 9, 751–767. [Google Scholar] [CrossRef] [PubMed] - Hudder, A.; Nathanson, L.; Deutscher, P. Organization of Mammalian Cytoplasm. Mol. Cell. Biol.
**2003**, 23, 9318–9326. [Google Scholar] [CrossRef] [PubMed] - Kissick, D.J.; Muir, R.D.; Simpson, G.J. Statistical treatment of photon/electron counting; extending the linear dynamic range from the dark count rate to saturation. Anal. Chem.
**2010**, 82, 10129–10134. [Google Scholar] [CrossRef] [PubMed] - Evans, R.J.; Boersma, J. The Entropy of a Poisson Distribution. SIAM Rev.
**2006**, 30, 314–317. [Google Scholar] [CrossRef] - Moore, C.C. Ergodic theorem, ergodic theory, and statistical mechanics. Proc. Natl. Acad. Sci. USA
**2015**, 112, 1907–1911. [Google Scholar] [CrossRef] [PubMed]

**Figure 1.**Normal lymphocyte (

**a**) and Jurkat cell (

**b**) in the PW matrix. At about the center of each cell, 50 × 50 pixel ROIs (5.5 µm wide) are marked.

**Figure 2.**Live cell time-dependent GLIE25 of one typical measurement unit and its Fourier analysis. Images were acquired at 0.5 s intervals out of which GLIE25 values were calculated (

**a**), and its power spectrum was analyzed (

**b**).

**Figure 3.**Schematic representation of an intracellular element (yellow circle) positioned between a light source (not shown) and a light detector (single pixel camera). The detecting space is represented by the light brown rectangle. Three representative positions of the element in relation to the detector’s detecting area are as follows: First, the centers of detection area and particle coincide so that the detecting pixel is maximally shadowed, resulting in the lowest gray value. Second, the element is on the right and out of the detection area, resulting in the maximum gray value. Third, the element is positioned to the left of the origin, only partially concealing the detector, resulting in an intermediate gray value. Each Gaussian represents the particle translocation distribution for different diffusivity in a time lag of 1 s (see inset).

**Figure 4.**Schematic representation of Equation (3): Entropy versus diffusivity in a time lag of 1 s (matching our experimental frequency range).

**Figure 5.**Schematic diagram of the positive correlations between diffusivity and entropy of intracellular particles, and diffusivity and related GLIE fluctuation-based measure: HFAP.

**Figure 6.**The dependency of number of gray values, AGLIE and derived fluctuate measures on light intensity. The parameters that were measured in different light intensities in the 5 × 5 pixel measurement units of the 6 measured areas were: (

**a**) Average number of gray values, (

**b**) the SD of the number of gray values in the time dependent 200 measurements, (

**c**) the average GLIE values and (

**d**) the SD of GLIE values in the time dependent 200 measurements. Points are connected by a curve in order to emphasize the trend of the results. Coefficient of variation never exceeded 1%.

**Figure 7.**Average values of AGLIE and HFAP parameters in the three groups of cells: dead Jurkat and dead lymphocyte cells (black), normal live lymphocytes (blue) and malignant lymphocytes—Jurkat cells (red). The ellipses around the average values in the cell group represent one SD of AGLIE and HFAP values.

**Table 1.**Results of AGLIE and HFAP (average and SD) obtained in dead Jurkat cells, live Jurkat cells, live normal lymphocytes, and dead lymphocytes. The p-values are located between two relevant columns.

Parameter | Dead Jurkat | p-Value | Jurkat | p-Value | Normal Lymph | p-Value | Dead Lymph |
---|---|---|---|---|---|---|---|

# of cells | 32 | - | 43 | - | 60 | - | 75 |

AGLIE | 2.72 ± 0.08 | p < 10^{−5} | 2.15 ± 0.10 | p < 10^{−5} | 2.65 ± 0.08 | p < 10^{−5} | 2.7 ± 0.07 |

HFAP | 1.35 ± 0.05 | p < 10^{−5} | 1.68 ± 0.06 | p < 10^{−5} | 1.40 ± 0.06 | p < 10^{−5} | 1.34 ± 0.05 |

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**MDPI and ACS Style**

Wohl, I.; Zurgil, N.; Hakuk, Y.; Sobolev, M.; Deutsch, M.
Fluctuation of Information Entropy Measures in Cell Image. *Entropy* **2017**, *19*, 565.
https://doi.org/10.3390/e19100565

**AMA Style**

Wohl I, Zurgil N, Hakuk Y, Sobolev M, Deutsch M.
Fluctuation of Information Entropy Measures in Cell Image. *Entropy*. 2017; 19(10):565.
https://doi.org/10.3390/e19100565

**Chicago/Turabian Style**

Wohl, Ishay, Naomi Zurgil, Yaron Hakuk, Maria Sobolev, and Mordechai Deutsch.
2017. "Fluctuation of Information Entropy Measures in Cell Image" *Entropy* 19, no. 10: 565.
https://doi.org/10.3390/e19100565