Simplified modeling of E. coli mortality after genome damage induced by UV-C light exposure

UV light is a group of high-energy waves from the electromagnetic spectrum. There are three types of UV radiations: UV-A, -B and -C. UV-C light are the highest in energy, but most are retained by the ozone layer. UV-A and -B reach the earth’s surface and cause damage on living organisms, being considered as mutagenic physical agents. Numerous test models are used to study UV mutagenicity; some include special lamps, cell cultures and mathematical modeling. Mercury lamps are affordable and useful sources of UV-C light due to their emission at near the maximum absorption peak of nucleic acids. E. coli cultures are widely used because they have DNA-damage and -repairing mechanisms fairly similar to humans. In here we present two simple models that describe UV-C light incidence on a genome matrix, using fundamental quantum–mechanical concepts and considering light as a particle with a discontinuous distribution. To test the accuracy of our equations, stationary phase cultures of several E. coli strains were exposed to UV-C light in 30 s-intervals. Surviving CFUs were counted and survival/mortality curves were constructed. These graphs adjusted with high goodness of fit to the regression predictions. Results were also analyzed using three main parameters: quantum yield, specific speed and time of mortality.

www.nature.com/scientificreports/ We used several variables for the mathematical modeling including the effective impact section (EIS), molecular excitation time, frequency of impact, lethal impact number (LIN), number of bases by unit of genomic volume (N*), lethal radiation dosage at 50% (LRD 50 ), absolute death state, minimum mortality time (MMT), specific speed of mortality (SSM), and cell quantum mortality yield (cQMY). The genomic volume for each E. coli strain was also calculated using LIN and cQMY. The fitness of the developed equations was challenged on the laboratory. Experimental procedures. Table 1  CM0067, OXOID) until density reached a turbidity equivalent to a 0.5 McFarland standard or ≈ 10 8 cells. Then, 3 ml of each culture were directly poured on dry Petri dishes (area 75.42 cm 2 ) and were exposed to UV-C light using a low-pressure mercury lamp (TUV, input voltage 15 W and output 3 W, Cat. No. G15T8, PHILLIPS, Holland) with 253.7 nm in wavelength that was located at a fixed distance of 33 cm, hence a fluence of 2.75 mJ/s cm 2 . Lamp was pre-heated for 10 min before use. Ten dishes of each strain culture were exposed for different times, in 30 s-intervals, up to a maximum exposure lapse of 300 s (fluence: 83-830 mJ/cm 2 , dosage range: 0.627-6.27 J/ Petri dish). Subsequently, 500 µl of the content of each dish were collected, serially diluted on saline solution (five dilutions on a decimal scale). Final dilution was mixed with soft agar and spread over nutrient agar plates (Cat.No. OX-CM0003B, OXOID) before incubation at 37 °C during 24 h. Control were obtained as experimental mock-ups. Briefly, 3 ml of the initial stationary culture were poured on dry plates, recovered, serially diluted and cultured under the same conditions as the experimental samples. The number of colony forming units per milliliter (CFU/ml) was determined with a colony counter (Accu-Lite Colony Counter Model 133-8,002, FISHER) and accounting the dilution grade. The number of CFUs from unexposed cultures was considered as 100% of viability. Three experiments were executed by triplicate for each exposure time.
Statistical analysis. Data on viable counts were processed using a linear regression analysis and calculating the correlation and determination coefficients, fiducial limits, confidence intervals, and finally subjected to a t-Student test by one mean. We report mean values of data from each cultured strain.

Results
Construction of the models. Physical model. Our physical model assumes that the occurrence of photoreactions depend on the number of photons impacting each nitrogenous base, and hence, on the energetic content of the radiation 25,28,35 . From the quantum perspective is expected, first, that an "activation" of the genome occurs when hitted by a luminous radiation, meaning that part of the incident energy has been absorbed but that is still not enough to produce bond breakage. Second, that the energy of the activated molecule increases as the wavelength diminishes; and third, that an activated molecule absorbs a fixed number of quanta to become an excited molecule where energy is suffice to provoke the dislocation of electrons (Fig. 1). If each excited molecule then results in a product molecule, the yield of such reaction will have values under one because it is a multiquantum process, therefore, the absorbed light influences the rate of cell death in a wavelength-dependent manner 36,37 . According to this, for any genomic matrix undergoing a photochemical reaction, each molecule accepts luminous energy independently instead of being evenly distributed among the whole bases.
We defined mortality as the condition were a cell is non-reproductive and has no metabolic activity and the cell quantum mortality yield, or cQMY ( µ ), as the capacity of quanta with a particular wavelength to cause chemical changes in sensitive bases. In a condition where the stimulus is continuous, each dead cell is the result of the sum of the alterations in the bases (or number of pyrimidine dimers), so cQMY can be expressed in terms of absorbed Einsteins as 37 : According to this, the rate of cell death or specific speed of mortality (SSM or k = dN/dt) would be proportional to the activation energy or radiation dosage:  www.nature.com/scientificreports/ When these expressions are multiplied by a proportionality constant and rearranged, a first order and first grade differential equation is obtained: also: cQMY is actually the proportionality factor defined as the efficiency by which a luminous stimulus changes the cellular mortality. dN dt corresponds to SSM, ΔE is the activation energy (also E a ), t is the exposure time and N is the number of dead cells. In other words, cQMY corresponds to the number of dead cells due to the formation of dimers in a time unit divided by the number of absorbed Einsteins.
Integrating Eq. (2) and assuming that Ф μ is an unchanging and cell-specific value, we have the following mortality expression: Considering that after UV light exposure, the number of initial cells (N 0 ) is given by the sum of dead plus surviving cells (N 1 ), then N 0 = N + N 1 and survival is defined by the equation: where N 1 < N 0 is always true. From here, E a can be expressed in terms of wavelength (λ) accordingly with assumption 4, so: The latter equation shows that specific speed of mortality by UV-C light is a converse function of the wavelength and is directly related to the total activation energy.
Besides, depending on cQMY, the time for cell inactivation depends on the nature of the sensitive chemical groups on the genome, its base distribution, sequence of sensitive sites and the functionality of cellular repairing mechanisms 12,25,38 .
Noticeably, cQMY has a linear relationship according to Eq. (4a), through measurable variables (wavelength and mortality). Hence, it is possible to calculate cQMY by counting dead cells after in at least two exposure times. Another way to obtain cQMY is to construct a graphic of N versus ln(t) where the slope corresponds to: www.nature.com/scientificreports/ On the other hand, when N 1 is charted versus ln(t), the expected curve is a straight line passing through the ordinate-axis and moving up or down by changes in the negative slope. Such intersect corresponds to N 0 . The slope of this survival curve is associated to the target size, this is to say, to the size of the genome at a constant repairing time. In the case where values of cQMY are the same for two microorganisms, its ratio to the relative genome sizes could be calculated from each slope, thus allowing the differentiation between species (Fig. 2). However, � µ changes with the repairing capacities of each cell and as a function of the degree of damage 13,39 . It is also worth noting that microorganisms with the same quantity of DNA and similar pyrimidine/purine ratios might have different sensitivities to the lethal effects of radiation depending on its genomic sequence; therefore each species have unique mortality times.
From Fig. 2, it can be observed that the intersection point between the mortality and survival curves corresponds to an equilibrium state where the number of dead is the same as the surviving cells (N = N 1 ), this point is also known as the Lethal Dose of Radiation (LDR 50 ).
Lethal dose of radiation. LDR 50 is defined as the exposure time needed to kill half a microbial population. Given Eq. (3) and considering that at LDR 50 N = N 0 2 , the intersection point using Eq. (4) is resolved as: In the special case where the population is only two microbial cells irradiated (N 0 = 2): from here we know that ln(t) = 1 k and this value corresponds to the minimal exposure period required to produce one dead cell, also known as Minimum Mortality Time (MMT, t u ) : Minimum time of mortality. The energy recquired to produce the death of one cell is provided by the incidence of UV-C light quanta on each nitrogenous base, instead of the uniform distribution of total energy on a wide genomic area that is contemplated by the wave-like perspective of the diffusion of light over matter (assumption 2).
Cell inactivation occurs in the same time-lapse at which photoreactions are happening. This brief period is such that the sum of the times of each individual photon-base collision is equal to t u , accounting that the repairing mechanisms are not fast enough to contain the damage.
On the other hand, we can express Eq. (6) on terms of wavelength as: the latter equation also shows that the variation on the time for cell death on each species is directly proportional to the wavelength and conversely proportional to cQMY. It also can be observed that MMT is defined by the combination of three fundamental constants of physics with cQMY, so both variables could be considered as intrinsic biological constants, associated to the nature and composition of a genome. MMT is a crucial concept for the development of our model and deserves special description. Mortality is frequently a polemic issue because cell death does not happen instantly and it can be described by a series of biochemical events and photochemical reactions. In here we adopted a simplified approach where MMT coincides with the formation of pyrimidine dimers in a context of insufficient repairing systems, the latter is considered as cellular inactivation. www.nature.com/scientificreports/ Absolute death state. The absolute death of a microorganism population is such that the number of surviving cells must be N 1 = 0, under such premise, according to Eq. (4) the time required to reach this state (T) could be expressed as: Mathemathical model. Analytic approach. The physical model describes the survial and mortality kinetics for entire cellular populations. We then aim to mathematically derive equations from molecular events as the initial point and deducting the behavior of individual cells and populations. In here, the total sum of accumulated photon-base collitions determining the death of a single cell is called unitary mortality process. The photodynamics of this process obliges to define the properties of cell populations in function of discrete variables with fixed values contained on discrete parameters as listed on Tables 2 and 3.
Mean free path. In here, we used an statistical mechanical concept known as free path 40 to define the trajectory that a photon follows from the moment of its impact on the genome until its absorption by a nitrogenous base. We assume that during its displacement, a photon moves at the speed of light (c) in a straight line and the mean distance of these motions is called mean free path (MFP). For photons passing without impacting any molecule, MFP would be infinite. For molecules with a big area without internal movement of photons, MFP would be zero. Actually, every nitrogenous base has a finite extension, then MFP is related to the size of the target, to the genome density and to the speed of impact. Since the photon is considered as a particle with rapid movement, it interacts with the target molecule during a short interval and the impact section depends on the speed (or energy).  www.nature.com/scientificreports/ To calculate the MFP in a simple manner, an area around the target could be constructed such that an absorption occurs if the trajectory of an impinging photon passes through it. This area is known as Effective Impact Section (EIS) and is represented by σ. A photon flux in a time t could traverse a volumen with a σ of ct and photons would collide with all the molecules with centers included in such volumen. For a continuos flow, the first photon covers a distance c t 1 finishing at the moment of its impact, a second photon travels c t 2 , a third c t 3 , and so on, in such way that the total distance is: where each time interval has a unique value.
If there are N * bases in a genomic volume ( V G ), the number of collisions at the time t is the number of remaining available bases ( 2N j ) in the cross-section σ. This is the probability of occurrence of an impact, a.k.a. absorbed fraction ( δ): The mean free path ( l ) is the average distance between impacts and then X i is the total distance at the total time invested on photonic impacts. The latter expression is equivalent to the genome thickness divided by the number of impacts occurring in the lapse t i : In order to find the average impact time, we considered that the time between absorption events is: The latter equation represents the excitation time 37,41 , which is an even briefer lapse than MMT. Thus the impact frequency or number of impacts in a time unit results from the reciprocal of Eq. (8): To obtain these equations we supposed that photons reach immovable targets. On the other hand, the collision process for a photon flux could be considered as elastic collisions because several particles moving at the same speed with different paths reaching one base would made the times of consecutive hits of single photons on multiple adjacent bases.
The term (N * σ ) is known as effective macroscopic section (EMS) and is represented by symbol Z. Considering the relationship between the time a photon requires to pass through the entire genome and the mean absorption If each base has a defined EIS, the total impact area is: σ (N * A G �x) , henceforth the probability of photon impacting the genome is a ratio of the total area or: σ (N * A G �x)/A G . To determine such ratio, we matched the equation to the absorbed fraction: Since x = c t and considering that N * = 2N j /V G , then substituting on Eq. (10) we have: www.nature.com/scientificreports/ Also, if A G x is the volume occupied by the genome, then A G �x/V G = 1 and the absorbed fraction is: An equivalent expression is: Taking a mean genome size of 5.2 millions of base pairs (Mbp) 42 for E. coli, a genome volume of 4.3716 * 10 −13 cm 3 and a thickness of x = 5.5111 * 10 −5 cm 43-45 , then substituting in Eq. (10): Lethal impact number. From Eq. (9), the product of the impact frequency times MMT is equal to the number of impacts at the time t = t u , also denominated lethal impact number (LIN) or the number of photonic impacts received by a particular molecule in the time needed to reach mortality, and it is proportional to the number of bases in the genome (see next section). Then: And accordingly, where it can be seen that k is directly proportional to the number of bases in the genome volume, and also, to the EIS. Generalization of the Unitarian equation system for mortality and impact. The frequency of impacts occurring in the time interval necessary for the death of one cell, gives an adequate argument to know the mortality kinetics, since the change rate in a population represents a continuous function in the close interval t a ≤ t ≤ t b and is derivable in the open interval t a < t < t b ; hence, for a value of t = (t u ) , enclosed between t a < t < t b , is verifiable that: then the interval between t a and t b constitutes a differential element of the mortality curve. Knowing that f(t u ) happens in t = (t u ) , then: rearranging the equation for t u = t b − t a and considering Eq. (14) we get the following expression: Substituting in Eq. (16), from N a = 0 until N = N b = 1 and for the death of a single cell as a condition of Eq. (15) with t a = 1, we have: This last expression is the definition of the SSM. Solving the differential equation we get the following equality for the mortality: On the other hand, the number of surviving microorganisms is:   (17) and (18)  where V G is the average of genome volume [44][45][46][47] . Substituting the known values for E. coli, we get: If MMT is the reciprocal of k, then we have a mean value of: In the molecular context, the energy distribution in the excited bases or "hot bases" is widely enriched as compared to the molecules in basal states; this is, electronically, vibrationally and rotationally. According to our estimations, the excited molecules will have a half-life of 10 −8 s 37,41 , and its energy would be either transferred to the adjacent pyrimidine to form a dimer or released with the consequent recovery of the basal state. This process occurs in the whole genome matrix, and could inactivate the cell if the UV-C light stimuli are constant and the repairing mechanisms are not sufficient.
The mean cQMY was calculated from Eqs. (13a) and (3a): Using the values of the proportionality hypothesis and the result for the SSM, we can calculate the cQMY as: Experimental results. We statistically analyzed the data obtained in the laboratory to graph the natural logarithm of exposure time and the mortality/survival of the exposed population. To simultaneously proof the results obtained through the models, regression coefficients and the intercepts were determined. The first step was to elaborate a plot using the CFU/ml counted for each strain to decide which was the best model describing the survival of E. coli under UV-C light exposure at several times ( Fig. 2A). Data were distributed in a logarithmic curve (survival decreases with exposure time), with N 0 and k as constants for the initial number of microorganisms ( t = 0 ). We use a regression model for the best fit, to establish the relationship between the variables and to check the theoretical prediction obtained from the mathematical models.
It can be seen from the Fig. 3A that data had a trend towards a linear behavior, with a slope or regression coefficient corresponding to the SSM. After constructing the confidence interval, it can be seen that 90% of the data were included inside the ranges (Fig. 3B).
Then, cQMYs were estimated using the regression coefficients (Fig. 4). It is also shown a complementarity between the variables, for instance, polAstrain (light green bar) had the highest SSM meaning that it dies with a shorter exposure time (lower MMT) and it had a high susceptibility to UV-C light exposure (high cQMY).  Proportionality hypothesis. The quantity and distribution of the nitrogenous bases and the genomic volume are intrinsic features of each genus and species and these parameters are determinant when calculating the SSM, because it changes inversely with V G . In here, we hypothesized that the number of photons impacting a genomic matrix is proportional to the number of base pairs and that the coefficient of these two values must be nearly equal to one 23,28 .
Then, using Eqs. (2a), (13) and (13a), we derived an equation to test the validity of the proportionality hypothesis: In some of our experimental observations this coefficient resulted in values close to one, except for W3110 and polA + strains. To compare other proportionality coefficients, Eq. (22) was applied to data previously reported in the literature (Table 4).
Nonetheless, Eq. (22) does not describe the number of photons necessary for the formation of pyrimidine dimers; to that end, we obtained the following expressions:  www.nature.com/scientificreports/ where D p is the number of formed pyrimidine dimers, D p is the number of photons necessary for the formation of a single dimer and γ abs is the number of absorbed photons per cell. The reciprocal values of cQMY are shown in Table 4, confirming postulate 7 and the multi-impact nature of UV-C light propagation on DNA.

Discussion
According to our equation system, the determination of cQMY, SSM, MMT and D p are crucial when describing UV-C light incidence on a genome. These four parameters are sensible indicators of the degree of damage of an exposed genomic matrix. Beggs 23 reported that Φ µ might be ranging from 10 -6 to 10 -5 and in Table 4 we enlisted other cQMYs 12,26,39,48 . Noticeably, the yields calculated experimentally are on the same order of magnitude, showing a high level of fitness in our equations. The proportionality coefficient of Eq. (22), varies from 1.03 to 4.00 (Table 4), we attribute these discrepancies to the diversity of methodologies used by each author.
The mathematical model predicted a value of SSM of 6.8 * 10 7 s -1 and the experimental results show a range of 2-7 * 10 7 s -1 , without a significant difference between both values (p = 0.045). In regard to MMT, the experimental results are within the range of the predicted value (1.4 * 10 -8 s) (Fig. 3) and coincide with other mortality times previously reported for the duration of the excitation state. We determined that photons could lead a base into its excitation state at a rather fast time (2.8252 * 10 -15 s, λ = 253.7 nm), which is also in accordance with other reports 35 .
Another useful data in the simulation of the process is the absorbed fraction ( δ ). In our experiments, δ = 0.65 , implying that approximately 35% of an E. coli genome is not hit.
Variations on this value are due to the genomic area, base-distribution and sequence configuration of each strain.
We used 5.2 Mbp as the mean genomic size according to the GENOME database for E. coli 42 , although other authors use 4.6 Mbp as the mean size (4′639,221 bases) 13,49 , and even in the literature, values ranged from 4.6 to 5.5 Mbp; nonetheless, by substituting these alternative values the results are approximate to those previously reported.
In regard to the physical model, although single impact developments are used in other biological models of UV-inactivation, they do not completely explain the dynamics of the process 23 , nor does the models with a continuous light diffusion approach (undulatory behavior). We considered that the corpuscular approach is more adequate since a quantum model results useful to explain why several photons are required to cause a pyrimidic dimerization. Besides, the repairing mechanisms play an important role because a cell with a low repairing rate could not overcome dimer formation and it might be unable to maintain its integrity. cQMY is directly associated with the wavelength of the incident light and to the genome size, which make it an appropriate measure of the effectiveness of a photochemical reaction (Eq. 21) and could be considered as a biological constant, analogous to the quantum yield. Our results are mutually compatible with the unitary equations of mortality and impact and in agreement with the experimental values reported by Kumiko et al. of 4 * 10 -5 for a genome size of 4.6 Mpb and the 6.4 * 10 -5 reported by Howard-Flanders et al. 26,39,48 . Sensitivity. We showed that SSM changes directly with cQMY and conversely with MMT. Nonetheless, the variability of data between each E. coli strain is related to the characteristics of each genome and/or its repairing mechanisms. Pyrimidine dimers are eliminated by the nucleases coded in uvrABC and PolA genes. When the filling of gaps is produced after replication, DNA repairing could be done by recombination with an undamaged parallel strain by means of Rec proteins, particularly by RecA 50 . This protein is also necessary for the expression of the SOS regulon, because it promotes the proteolysis of the repressor (LexA) hence allowing the expression of umuCD genes 51 . On the other hand, it is known that deletion-insertion mutants of the DNA helicase II (uvrD) are UV-C light sensitive due to the high variability of its allelic genotypes 52 .
In here, the elimination genotypes of each strain produce different values for MMT, cQMY and SSM. The strain HfrH180, which does not have deficiencies on any DNA repairment system 53,54 was moderately sensitive to UV-C light effects.
PolA − strain presented a higher sensitivity to the UV-C light exposure compared to PolA + , probably due to its lack of the polymerase. In fact, the pair of PolA − /PolA + strains is widely used on the study of geno-lethal effects of diverse substances on bacterial test panels 55 .
Youngs et al. 56 showed that a higher sensitivity to UV light was presented by strains with a recA − genotype, being the strains recA − /uvrB-the most susceptible. In here, the strain DH5αthy − was less sensitive to UV-C light than HfrH180 although it does not have mutations on the uvrABC system. The ATCC25922 strain had a relatively high sensitivity only below polA − , which was unexpected, although, it partially could be explained because this strain is intrinsically sensitive to lethal agents. In fact, ATCC25922 is recommended by the CLSI as a reference microorganism to evaluate antimicrobial sensitivity. Finally, E. coli W3110, previously reported as UV-sensitive and with a pyrimidine auxotrophy, had low SSM but a still measurable mortality.
Regression, correlation and determination. The regression analysis showed a strong trend towards a straight line on the dispersion diagram [ln(t) vs N, Fig. 2B] and the regression coefficients were in agreement with the theoretical calculations (SSM, k). The fact that the data behavior was not accurately predicted for a sample of the variables [(ln(t i ),N j ) i,j ] depends on the experimental variability and the complex biological nature of test system. Let us also recall that we assume that the number of surviving or dead microorganisms is a random variable, which depends on the exposure time, that it has homogeneous variances, independence of N and N 1 (24) D p = 1/� µ = γ abs /D ∧ p Scientific RepoRtS | (2020) 10:11240 | https://doi.org/10.1038/s41598-020-67838-1 www.nature.com/scientificreports/ values and a normal distribution. The proposed models are adequate to describe the travel of UV-C light in a genomic environment in a more complete and real manner. Regarding the correlation coefficients, the points on each strain showed a marked association between both variables [ln(t), N] within the interval 0.9526 ≤ R ≤ 0.9949. Hence, there is good evidence that the variable ln(t) contributes to explaining N and N 1 .
The values for the determination coefficients had a high goodness of fit to the projected curve because the explained fractions are in the interval of 0.9074 ≤ R 2 ≤ 0.9898, revealing a cause-effect relationship between the variables.

Conclusions
The quantum depiction proposed in these models was useful to prove that the distribution of the activation energy from UV-C light through the genome is discontinuous. On the other hand, these are multiple-impact mechanistic models that allowed the calculation of mean population values from individual effects at the photonmolecule interaction. Finally, other population kinetics of mortality and survival could be simulated by taking into account the wavelength of the lamp used, the volume and base distribution in the genome of the microorganism in study and by contrasting data with simple viable counts measures from the resulting cultures.