# Biophotons: A Hard Problem
## 2 Methods and Experimental Data
The time scales relating to the single bean and the 76 lentil seeds are completely different; therefore, to highlight the common characteristics of the two emissions, we rescaled the time scale of the bean by a factor of 0.164. This allowed us to align the two maxima, the C peaks in the figure. The zero was placed in the first minimum of both emissions at the end of the residual luminescence; this means we subtracted the values 10 and 100 from the original time scales, respectively, for the lentils and the single bean. Furthermore, to normalize the number of counts to the C peaks, we multiplied the counting scale of a single bean by a factor 2.28. The time-rescaling procedure used here is based on the popular logistic equation [28; 29], which can be used in different systems to describe the growth of a population that reaches the final steady-state value that is specific for any system. The logistic equation plays an important role in biology, also contributing to the emerging science of chaos. Here, the logistic equation takes the form  
\[\dot{n}(t)=a\cdot n(t)-b\cdot n^{2}(t) \tag{1}\]  
where \(n(t)\) is the number of cells growing because of watering the seeds [29], and the numbers a and b are constants that depend on the system. The solution to Equation (1) can be written as:  
\[n(t)=\frac{a\cdot Ce^{at}}{1+b\cdot Ce^{at}} \tag{2}\]  
where C depends on the initial conditions \(n(0)\) through the relationship \(C=\frac{n(0)}{a-n(0)\cdot\beta}\). We made the conjecture that the rate of biophoton emission is proportional to the derivative of the number of cells, i.e., to \(\dot{n}(t)\):  
\[\dot{n}(t)=\frac{a^{2}\cdot Ce^{at}}{\left(1+b\cdot Ce^{at}\right)^{2}} \tag{3}\]  
Cells can be thought of as a kind of interacting unit in the living organism; for a single type of unit, the time derivative reaches a maximum at a time determined by the parameters and b and the initial conditions. The corresponding emission has a regular trend, with only one maximum reached at time \(t_{max}=\frac{1}{a}\text{ln}\Big{(}\frac{1}{\beta}\Big{)}\) and intensity \(I_{max}=\frac{a}{4\beta}\). \(\beta\) is defined as \(\beta=b\cdot C\). We hypothesized that the saturation time of the ordinary logistic equation in different systems corresponds to the maximum emission peak, and we rescaled the time scale of the bean so that its maximum photon emission rate coincided with that of the lentils. In this way, we made a comparison between them, reducing the time of the slower growth by a reducing factor: 0.164 in the case of the bean. The fact that the emissions of different seeds have very similar temporal behavior led us to hypothesize the existence of a sort of generalized logistic equation as a universal property of the connection between the system growth and photon emission. The experimental data show a wealth of structures with a succession of maxima and minima distributed throughout the experiment. In detail, the lentil emission shows peaks B and C, separated by about 5 h, between time 0 and 20 h, and the same two peaks (B' and C) are present in the emission of the single bean, but here they separated by about 14 h. The biophoton emission of the bean has a further peak A at about 43 h (these values in the bean time scale), which is absent in the lentil emission. Both emissions also show at least two slopes in the growth phase, between zero and peak C.  
In our opinion, this is clear evidence of the presence of different types of units in the seeds that could be activated at different times on different time scales. The simplest generalization of Equation (3) that considers the presence of different types of units and with which to make a phenomenological fit of biophotonic emission as a function of time could be of the following type:  
\[\dot{n}(t)=a^{2}e^{at}\sum_{i=1}^{J}\frac{C_{i}}{\left(1+b_{i}\cdot C_{i}e^{at }\right)^{2}} \tag{4}\]  
where the different constants could be determined by a fit procedure conducted by using the experimental data.