## 3 Results  
### Gray Level Information Entropy (GLIE) Is Sensitive to the Motion of Intracellular Particles  
Information entropy, \(E\), can be calculated for the intensity distribution across the pixels of an image. Such entropy expresses the homogeneity of the inspected medium. The scale of changes in entropy is logarithmic due to the entropy function (\(E=-\sum P_{i}Ln(P_{i})\), where \(P_{i}\) is the probability of occurrence of an intensity, \(i\), and summation is over all intensity values). Here, we focus on microscopy imaging of the interior of a cell in order to characterize the motion of intracellular particles. Such motion may produce pronounced fluctuations in the homogeneity and information entropy obtained from the intensity of relevant pixels. Such fluctuations depend on the size of the particles, their diffusivity and the pixel size. For illustrations, we provide an example of a simple case of a particle undergoing a random walk in a homogeneous medium. The particle is captured by five pixels, as illustrated in Figure 1. In this situation, the particle motion will influence the readout intensity of specific pixels as follows. When the particle is located at the center (Figure 1a), the five intensities of the pixels will produce only two different gray levels. Accordingly, the information entropy calculated for the readout of the five pixels, is \(E=1.43\). When the particle moves about 0.25 pixel-length from the center (Figure 1b), the intensity of the five pixels will now produce four different gray values, and \(E=5.47\). When the particle moves about 0.5 pixel-length from the center (Figure 1c), the five intensities of the five pixels will produce now three different gray levels, and \(E=3.44\). Thus, in this example the information entropy value of the five pixels (\(E\)) is fluctuating considerably between 1.43 and 5.74 due to the particle's motion. Next we consider a more realistic and complicated example of the motion of an intracellular particle (Figure 2). Here, the pixel width is taken as 0.16 \(\upmu\)m (as in our experimental system) and the particle diameter is considered to be \(\sim\)0.48 \(\upmu\)m. This is a common (diffraction-limited) size of small intracellular entities, such as vesicles or sub-micron organelles [30]. Diffusivity of intracellular vesicles was measured to be 0.012 \(\upmu\)m\({}^{2}\)/sec in normal cells (i.e., Jurkat cells with their normal metabolic activity) and 0.001 \(\upmu\)m\({}^{2}\)/sec in ATP-depleted cells (with no mechanical work and only thermal agitation) [4]. We next consider a time lag of 0.2 s for inspecting the particle motion (as the time lag used in our experimental setup, which fits the time scale for intracellular active diffusion \(>\) 0.1 s). Accordingly, for a single time lag, the standard deviation (SD) of the Probability Density Function (PDF) of translocations  
Figure 1: An example of a particle motion captured by five pixels in a homogenous medium. (**a**) The particle is located in the center, covers the length of the three central pixels. Accordingly, the five pixels present two gray levels and information entropy of 1.43. (**b**) The particle moves a length of 0.25 pixel to the right. In this situation, it fully covers the length of 2 pixels, partially and differently covers the length of two other pixels and does not cover one pixel. Accordingly, the five pixels present four gray levels and information entropy of 5.74. (**c**) The particle moves a length of 0.5 pixel to the right. In that situation, it fully covers the length of two pixels, partially and similarly covers the length of two other pixels and does not cover one pixel. Accordingly, the five pixels present three gray levels and information entropy of 3.44.  
of the particle is 0.07 \(\upmu\)m in a normal cell and 0.03 \(\upmu\)m for a particle in a cell after ATP-depletion.