## 3 Results
### Gray Level Information Entropy (GLIE) Is Sensitive to the Motion of Intracellular Particles
We consider a translocation of 0.4-0.6 of a pixel length (here, 0.064-0.096 \(\upmu\)m) in a normal active cell, for which the SD of translocation is 0.07 \(\upmu\)m. In contrast, in an ATP-depleted cell the SD of translocations is 0.03 \(\upmu\)m. This yields estimated probabilities of 0.095 for a normal active cell, in comparison to 0.015 for an ATP-depleted cell (see detailed calculations in Supplementary Materials Table S1). These probabilities reflect the higher occurrence rate of larger fluctuation in \(E\) due to the particle translocations in an active cell, relative to an ATP-depleted cell. Large non-equilibrium translocations of the particle that takes place in active cells and the corresponding large change in \(E\) (Figure 2c) are not expected to occur in ATP-depleted cells. Accordingly, we expect that fluctuations in \(E\) will be more pronounced in normal, active cells relative to ATP-depleted cells. Thus, parameters based on fluctuations in \(E\) of specific pixels could be utilized to detect and study intracellular mechanical activity. Intracellular liquid-liquid phase separation is a process by which specific cellular material form pronounced complexes, such as nucleoli, condensates, etc. [31; 32]. In mechanically active cells, intracellular liquid-liquid phase separation is reduced [6; 33]. This may facilitate higher homogeneity of the intracellular content. More homogeny in the intracellular medium will result in average lower values of \(E\). In this case, the fluctuations in \(E\) tend to be more pronounced due to the logarithmic nature of the entropy function (Figure 3). Therefore, in cells that are more mechanically active, fluctuations of \(E\) will be higher due to a synergic effect: an increase in the dynamics of intracellular particles, combined with an increase in the homogeneity of the intracellular content. Comparing Pixel Gray-Levels Information Entropy (GLIE) Fluctuations and Spatial Image Correlation Spectroscopy (SICS) Fluctuations in Live Jurkat Cells before and after ATP Depletion  
In the following section we test the ability of the proposed GLIE analysis to detect non-equilibrium mechanical activity in live cells. This is compared to the results in these cells after ATP depletion. The results of cell dynamics from GLIE fluctuation analysis were further compared to the results of cell dynamics, using the established tool of ICS analysis [29]. ICS analysis is based on analyzing spatial and temporal correlations of pixel intensities in a time series of images. When the spatial correlations (SICS) of each of the time-dependent images are also correlated in time, the dynamics of the related structures could be elucidated [29]. We studied fluctuations of GLIE or SICS by Amplitude Spectral Density (ASD) analysis and a power fit of the ASD results. We conducted live cell imaging of Jurkat T lymphocytes. The label-free cells were imaged 200 times with a 0.2 s interval. We studied a region of interest (ROI) of \(50\times 50\) pixels at the center of each measured cell, and conducted GLIE and SICS fluctuations analysis within (Figure 4a). For SICS analyses, results of the six lowest spatial lags from the horizontal and from the vertical axes were excluded because they were significantly influenced by PSF-dependent correlations. For the remaining 18 lags of each dimension, we conducted DFT analysis for each lag of the time-dependent correlations. The calculated DFT amplitudes of the \(18\times 2\) lags were then averaged to get the ASD results of that cell and also the related power fit parameters of that spectrum. The average ASD results from the SICS analysis of all the cells under each condition (before and after ATP depletion) are presented in Figure 4b. Figure 3: Gray Level Information Entropy (GLIE) values in a \(5\times 5\) pixel unit according to the number of gray levels in that unit. GLIE values were calculated based on a uniform distribution of the different gray levels in the 25-pixel unit or according to a Gaussian distribution of the different gray levels in that unit.