Keywords: data modeling, co-exposure, action spectra, non-additive effects, DNA damage

Photobiological effects are known to greatly depend on the wavelength of the incident photons that define the nature of the activated chromophores. A growing number of experimental data show that considering the effect of complex light sources as a sum of the effects of monochromatic exposures can be misleading. Indeed, the combined exposure to several wavelength ranges may modulate photobiological responses or even induce novel processes.

These observations are similar to a well-known topic in chemical toxicology: the non-additivity of effects in mixtures where either antagonism or synergy are often observed. In the present work, we investigated if a data analysis tool first developed for studying non-additivity in mixtures of drugs, the combination index, could be applied to photobiological processes. We chose to work on the formation of UV-induced DNA photoproducts where additive, antagonist and synergistic effects take place simultaneously. In addition to this application, we worked on the mathematical bases of the concept in order to broaden its applicability to phenomena exhibiting various dose-response patterns. We also addressed the question of the evaluation of the error on the determination of the combination index.

Synergistic or Antagonist Effects of Different UV Ranges

Analyzed by the Combination Index: Application to DNA Photoproducts Thierry Douki* and Arnaud Buhot Univ. Grenoble Alpes, CEA, CNRS, IRIG, SyMMES, F-38000 Grenoble Corresponding authors: Thierry Douki Email: thierry.douki@cea.fr Combination index (CI) is a data analysis tool reflecting antagonistic or inhibitory effects of complex exposures. Applied to DNA photochemistry, CI could unambiguously demonstrate the additive effects of UVB and UVA on the formation of cyclobutane pyrimidine dimers (CPD). CI also clearly assessed the enhanced yield of Dewar valence isomers (Dewar) at the expense of that of (6-4) photoproducts (64PP) upon exposure to combinations of UVB and UVA. These examples illustrate the usefulness of the CI strategy in photobiology.

INTRODUCTION

The activation and the intensity of photobiological responses are obviously dependent on the fluence of the radiation but also of the wavelength of the incident photons. This has been realized for a long time and is classically taken into account by the use of action spectra [START_REF] Björn | Action spectroscopy in biology[END_REF][START_REF] Sliney | Photobiological measurements and obtaining action spectra[END_REF][START_REF] Setlow | Action spectroscopy[END_REF][START_REF] Schäfer | Action spectroscopy[END_REF], which report the wavelength-dependent variation of the intensity of an effect per dose unit. By convolution with the emission spectrum of a source of interest, action spectra provide an estimation of the biological effect under specific experimental conditions. However, the use of action spectra is based on an additivity principle that is not always relevant in photobiology. Some photobiological effects may result from the consecutive impact of two different wavelength ranges. In this case, a synergy is observed. Conversely, there are cases where the response triggered by a wavelength range is decreased by another, leading to an antagonism.

Examples can be found in the photomorphogenesis of plants that is activated and inhibited at two different wavelengths, [START_REF] Withrow | Action Spectra of Photomorphogenic Induction and Its Photoinactivation[END_REF] or in bacteria where sunlight simultaneously induces DNA damage and activates photolyase repair enzymes [START_REF] Essen | Light-driven DNA repair by photolyases[END_REF][START_REF] Weber | Light-driven enzymatic catalysis of DNA repair: a review of recent biophysical studies on photolyase[END_REF]. As we recently reviewed, similar non-additive effects have been reported in humans and in mammalian cells [START_REF] Douki | Wavelengths and temporal effects on the response of mammalian cells to UV radiation: Limitations of action spectra illustrated by genotoxicity[END_REF].

Immunosuppression is induced by both UVB and UVA but UVA may counteract the effect of UVB [START_REF] Damian | An action spectrum for ultraviolet radiation-induced immunosuppression in humans[END_REF][START_REF] Garssen | UVA exposure affects UVB and cis-urocanic acid-induced systemic suppression of immune responses in Listeria monocytogenes-infected balb/c mice[END_REF][START_REF] Reeve | Ultraviolet A radiation (320-400 nm) protects hairless mice from immunosuppression induced by ultraviolet B radiation (280-320 nm) or cis-urocanic acid[END_REF]. Similarly, the extent of apoptosis depends on the ratio between the two wavelength ranges upon co-exposure to UVB and UVA [START_REF] Tuchinda | Comparison of broadband UVB, narrowband UVB, broadband UVA and UVA1 on activation of apoptotic pathways in human peripheral blood mononuclear cells[END_REF]. More recently, UVA was reported to modulate the cytotoxicity of UVB [START_REF] Ibuki | Long-wavelength UVA enhances UVB-induced cell death in cultured keratinocytes: DSB formation and suppressed survival pathway[END_REF]. Other examples concern the induction of oxidative stress. UVA was found to promote the formation of lipofuscin that is a visible light photosensitizer [START_REF] Tonolli | Lipofuscin generated by UVA turns keratinocytes photosensitive to visible light[END_REF].

Similarly, some UVB photoproducts of tryptophan like N-formylkynurenine are UVA-photosensitizers [START_REF] Dreaden | N-formylkynurenine as a marker of high light stress in photosynthesis[END_REF][START_REF] Park | The tryptophan-derived endogenous aryl hydrocarbon receptor ligand 6formylindolo[3,2-b]carbazole is a nanomolar UVA photosensitizer in epidermal keratinocytes[END_REF][START_REF] Walrant | N-formyl-kynurenine, a tryptophan photooxidation product, as a photodynamic sensitizer[END_REF]. Last, in the field of DNA damage, pyrimidine pyrimidone photoproducts (64PPs) are produced by UVB radiation in DNA and efficiently converted into their Dewar valence isomers (Dewars) by UVA photons [START_REF] Douki | Dewar valence isomers, the third type of environmentally relevant DNA photoproducts induced by solar radiation[END_REF][START_REF] Taylor | DNA, light, and Dewar pyrimidones: the structure and biological significance of TpT3[END_REF].

The non-additive effects of different wavelength ranges represent thus a key issue in the prediction and understanding of photobiological effects. Unfortunately, they are not frequently taken into consideration or only on a very limited number of applied doses. This last point is important because some effects can be additive within a dose range and antagonist or synergistic in another. The purpose of the present work is thus to provide a methodology for the identification of non-additivity effects in photobiology under the largest set of conditions possible with the most reasonable amount of experimental work. Ideas of experimental design and data treatment for this problem can be found in the field of the pharmaceutical and toxicological effects of chemicals, where examples of non-additive responses to the combination of organic or inorganic compounds are countless. The question of the non-additive effects has been put forward for many years with the notions of exposome and cocktail effects [START_REF] Escher | From the exposome to mechanistic understanding of chemical-induced adverse effects[END_REF]. Several mathematical models have been designed to analyze experimental data and establish the occurrence, or not, of synergy or antagonism in the impact of mixtures [START_REF] Lasch | Mixture effects of chemicals: The difficulty to choose appropriate mathematical models for appropriate conclusions[END_REF][START_REF] Olmstead | Toxicity assessment of environmentally relevant pollutant mixtures using a heuristic model[END_REF][START_REF] Rider | An integrated addition and interaction model for assessing toxicity of chemical mixtures[END_REF].

Among them, the combination index (CI) approach proposed by Chou [START_REF] Chou | Quantitative-Analysis of Dose-Effect Relationships -the Combined Effects of Multiple-Drugs or Enzyme-Inhibitors[END_REF][START_REF] Chou | Theoretical basis, experimental design, and computerized simulation of synergism and antagonism in drug combination studies[END_REF][START_REF] Chou | Preclinical versus clinical drug combination studies[END_REF][START_REF] Chou | The mass-action law based algorithm for cost-effective approach for cancer drug discovery and development[END_REF] is particularly attractive as it combines the a priori definition of an optimal experimental plan with a strict mathematical approach for the identification of antagonist or synergistic effect. This strategy was designed for the optimization of the efficiency of combinations of drugs. It was very successful in various fields. More than 1500 references to the combination index could be retrieved, for a large part in the field of cancer treatment but also in studies on HIV or in toxicology investigations. The most complete review on the combination index [START_REF] Chou | Theoretical basis, experimental design, and computerized simulation of synergism and antagonism in drug combination studies[END_REF] has received more than 4000 citations. The mathematical model designed by Chou is derived from the mass action law and is well suited to study the interaction of a drug with a physiological target [START_REF] Chou | Theoretical basis, experimental design, and computerized simulation of synergism and antagonism in drug combination studies[END_REF][START_REF] Chou | The mass-action law based algorithm for cost-effective approach for cancer drug discovery and development[END_REF]. Extensive mathematical analysis of the underlying phenomena led Chou and his co-workers to propose a rather simple equation describing pharmaco-kinetics interaction, the medium effect equation (MEE). The same mathematical framework led to the design of the CI, a convenient tool to study antagonist and synergistic effects.

As it originates from the mass action law, the MEE can precisely model data ranging between 0 and 1 (or 0 and 100 %) with dose-dependent curves of the responses exhibiting sigmoidal or hyperbolic shapes. In addition, the MEE requires only two fitting parameters thus limiting the required dose-response experiments for their determination. Determining the MEE is a requisite to the calculation of the CI. The value of the CI shows whether the effects of a mixture are additive (CI=1), antagonist (CI>1) or synergistic (CI<1). The strategy underlying the CI approach is illustrated on a theoretical example on Scheme 1. In the framework designed by In the present work, we aimed at providing a proof-of-concept that the CI approach could be used in photobiology. For this purpose, we applied the CI formalism to the well-known UV photochemistry of DNA (Scheme 2), and in particular to the formation of pyrimidine dimers [START_REF] Cadet | Formation of UV-induced DNA damage contributing to skin cancer development[END_REF][START_REF] Douki | Effect of denaturation on the photochemistry of pyrimidine bases in isolated DNA[END_REF][START_REF] Douki | Sunlight-induced DNA damage: molecular mechanisms and photoprotection strategies[END_REF]. These photoproducts arise, upon exposure to UVB, from the cycloaddition of the C5-C6 double bonds of adjacent pyrimidine bases or from the addition of the C5-C6 double bond of the 3'-end pyrimidine to the C4 carbonyl or imine group of the 5'-end pyrimidine. The respective products are cyclobutane pyrimidine dimers (CPDs) and 64PPs. CPDs are also produced by UVA, although in much lower amounts than by UVB. A specific feature of 64PPs is that their pyrimidone moiety can absorb a second UV photon and undergo photoisomerization into Dewars. In isolated photoproducts, this photoreaction is most efficient at approximately 320 nm [START_REF] Douki | Characterization of the (6-4) photoproduct of 2'-deoxycytidylyl-(3'-5')-thymidine and of its Dewar valence isomer[END_REF][START_REF] Taylor | DNA, light, and Dewar pyrimidones: the structure and biological significance of TpT3[END_REF][START_REF] Taylor | Quantitative conversion of the (6-4) photoproduct of TpdC to its Dewar valence isomer upon exposure to simulated sunlight[END_REF][START_REF] Lemaire | Quantum yields and secondary photoreactions of the photoproducts of dTpdT, dTpdC and dTpdU[END_REF], but this maximum is shifted towards the UVA region [START_REF] Douki | Dewar valence isomers, the third type of environmentally relevant DNA photoproducts induced by solar radiation[END_REF] within DNA because of the presence of overwhelming amounts of normal bases that shield UVB photons [START_REF] Douki | Relative contributions of UVB and UVA to the photoconversion of (6-4) photoproducts into their Dewar valence isomers[END_REF]. DNA photochemistry thus offers examples of additive (CPDs), synergistic (Dewars) and antagonist (64PPs) effects between UVB and UVA. In addition, the yield of photoproducts cannot be more than 100 %. The latter observation is reinforced by the fact that experiments are always performed at a low modification rate that mimics real life situations. In that respect, these data are well suited for application of the MEE. We also used them to explore the use of other modeling equations in place of the MEE. Indeed, the latter function does not reproduce all possible dose-course shapes of biological responses, which can be more complex than a drug / target interaction. In addition, numerous effects do not range between 0 and 1, and can exhibit no theoretical limit.

Scheme 2: Structure of the dimeric pyrimidine photoproducts induced in DNA by UVB and UVA. The photoreactions are described for two thymines but similar products can be formed at TC, CT and CC sequences. W UVA lamps used in the three first series while the 3 lamps were used in the last.

EXPERIMENTAL PART

1) A filter with cut-off at 320 nm (Schott, Colombes, France) was placed in front of the samples in the UVA beam. The fluences were constant and set at 0.13 kJ m -2 min -1 for UVB and 2.36 kJ m -2 min -1 for UVA. Exposure times ranged between 1 and 90 min, and the experiments were repeated twice.

2) The fluence of the UVA source (with a cut-off filter at 320 nm) was left constant at 0.65 kJ m -2 min -1 1 . The position of the UVB lamp (with a cut-off filter at 295 nm, Schott, Colombes, France) was changed to provide fluences of either 0.20, 0.16, 0.12, 0.09 or 0.03 kJ m -2 min -1 . Irradiations were performed once under each UVB conditions with exposure times ranging between 4 and 64 min. Samples were analyzed once except that receiving the largest dose which was analyzed three times. The values for this sample were averaged before data analysis.

3) UVB and UVA fluences (with cut-off filters at 295 and 320 nm, respectively) were kept constant at 0.08 and 0.73 kJ m -2 min -1 , respectively. Five irradiation protocols were used. DNA samples (2 mL) were irradiated for either 5, 10, 20 or 40 min. Three aliquot fractions (200 µL) were collected from each of these UV-exposed solutions. The three first protocols involved either UVA, UVB or UVB and UVA simultaneously. In the fourth design, samples were exposed for 5 min to pure UVA and then for 5 min to UVB.

The experiment was repeated with consecutive exposures to UVA and UVB lasting 10+10, 20+20 or 40+40 min. The fifth protocol was similar except that UVB was applied before UVA. All these experiments were referred to as UVA, UVB, UVB+UVA, UVAUVB and UVBUVA. Each irradiation was performed once but all collected samples were analyzed three times and the obtained results were averaged before data analysis.

4) The three lamps were used with bandpass filters of 310±5 nm for UVB, 340±13 nm for the 1×15 W UVA and 386±13.5 nm for 2×15W UVA. All bandpass filters were obtained from Edmund Optics (Lyon, France). The respective fluences were 0.0048, 0.036 and 0.045 kJ m -2 min -1 , respectively. Irradiation lasted 30, 60, 90 and 120 min.

The experiment was repeated three times.

Following irradiation, DNA was precipitated from the samples by addition of NaCl and cold ethanol. The level of photoproducts was then determined as previously reported [START_REF] Douki | The variety of UV-induced pyrimidine dimeric photoproducts in DNA as shown by chromatographic quantification methods[END_REF].

The DNA samples were enzymatically hydrolyzed by phosphodiesterases, DNases and alkaline phosphate. The resulting solutions were analyzed by high performance liquid chromatography coupled to tandem mass spectrometry.

DATA MODELING

All calculations were performed in Microsoft Excel spread sheets in order to copy and paste data directly from the software of the LC-MS/MS system. We determined the values of the parameters of the MEE (Dm and m), of the Dx and of CI using the method described by Chou [START_REF] Chou | Theoretical basis, experimental design, and computerized simulation of synergism and antagonism in drug combination studies[END_REF][START_REF] Chou | The mass-action law based algorithm for cost-effective approach for cancer drug discovery and development[END_REF]. We used Compusyn (ComboSyn Incorporated) on a subset of data to check that our calculations were correct. Excel sheets were also used for the calculation of the CI using other equations with the first set of experimental results. Values of the coefficients of the regressions and of their error were obtained from Origin (OriginLab Corporation, MA). For the calculation of the propagation of errors, we used the classical method stating that if a measure y is a function of a parameter x, then ∆y = |( df dx )| × ∆x, where ∆y and ∆x are the errors on y and x, respectively.

When y depends on more than one parameter, the absolute values of all errors are added:

∆y = |( df dx1 )| × ∆x1 + |( df dx2 )| × ∆x2 |( df dx3 )| × ∆x3 + ⋯

Medium effect equation

The MEE used in the original work by Chou et al. [START_REF] Chou | Theoretical basis, experimental design, and computerized simulation of synergism and antagonism in drug combination studies[END_REF] is the following:

(1)

𝑓𝑎 (1-𝑓𝑎) = ( 𝐷 𝐷𝑚 ) 𝑚
where fa is the response referred to as affected fraction in the framework of drug/target interactions, D the dose, Dm the dose leading to a 50% of the maximal effect, and m a coefficient reflecting the shape of the curve (m>1  hyperbolic; m=1  sigmoidal; m<1  "flat" sigmoidal).

The Dm and m parameters were determined from the experimental data by linear regression of

Log-Log curve (2) Log ( 𝑓𝑎 1-𝑓𝑎 ) = 𝑓(Log(𝐷)) = 𝑎 × Log(𝐷) + 𝑏 = 𝑚 × Log(𝐷) -𝑚 × Log(𝐷𝑚)
m is thus the slope a and Dm is 10 -𝑎 𝑏 .

Calculation of the CI requires the determination of the dose Dx of each factor leading to the same effect fx than a given mixture. In other words, it is necessary to invert the MEE equation

(1) to determine Dx for a measured effect fx, a known value of 𝑓𝑎 1-𝑓𝑎 :

(3

) 𝐷𝑥 = 𝐷𝑚 × [𝑓𝑥] 1/𝑚
The error on Dx (Dx) can be inferred from that on m and Dm. The coefficients m and Dm were obtained by linear regression of equation ( 2). Thus, m = a is the estimated error on the slope a and Dm can be shown to be:

(4) ∆𝐷𝑚 = (| 1 𝑚 | × ∆𝑏 + | 𝑏 𝑚 2 | × ∆𝑚)
The errorsDm and m can then be used to calculate the error on the determination of the dose Dx. When replicates of a same experiment are used, the mean 𝑓𝑥 is associated to the standard deviation 𝜎𝑓𝑥 used as an estimation of ∆𝑓𝑥:

(5) ∆𝐷𝑥 = |𝑓 𝑥 ̅̅̅ 1 𝑚 | × ∆𝐷𝑚 + | 𝐷𝑚×ln(𝑓 𝑥 ̅̅̅) 𝑚 2 × 𝑓 𝑥 ̅̅̅ 1 𝑚 | × ∆𝑚 + | 𝐷𝑚 𝑚 ̅ × 𝑓 𝑥 ̅̅̅ ( 1 𝑚 -1) | × ∆𝑓 𝑥
In the present work, fa is the level of DNA photoproducts. In contrast to most works in the field of chromatographic detection of DNA damage, results were not expressed in number of damage per million normal bases but in ratio of mol of damage per mol of DNA bases. The values are thus very small but range between 0 and 1 as necessary for the use of the MEE. In addition, the fa values used in the calculations with the MEE were normalized to the content of bipyrimidine sites at the origin of each photoproducts. For this purpose, we took into account the frequency of the different bipyrimidine dinucleotides in CT-DNA, which are 0.099, 0.061, 0.053 and 0.053 for TT, TC, CT or CC, respectively [START_REF] Matallana-Surget | Effect of the GC content of DNA on the distribution of UVB-induced bipyrimidine photoproducts[END_REF][START_REF] Douki | Contribution of cytosine-containing cyclobutane dimers to DNA damage produced by photosensitized triplet-triplet energy transfer[END_REF].

Polynomial fit

Calculation of the CI was also performed with polynomial fits of the DNA damage data. For this part of the work, results were expressed in number of photoproduct per million normal bases.

For Dewars, the experimental data were fitted using a second order polynomial equation.

Because no photoproduct is detected in non-exposed DNA, the origin was set to 0. The expression is thus the following:

(6) 𝑓𝑎 = 𝑎 × 𝐷 2 + 𝑏 × 𝐷
The dose Dx necessary for the induction of an effect fx is the positive solution of equation ( 6):

(7) 𝐷𝑥 = -𝑏+√𝑏 2 +4×𝑎×𝑓𝑥 ̅̅̅̅ 2×𝑎
with the error of Dx being:

(8) ∆𝐷𝑥 = | 𝑓𝑥 𝑎√𝑏 2 +4×𝑎×𝑓𝑥 - √𝑏 2 +4×𝑎×𝑓𝑥 2𝑎 2 | × ∆𝑎 + | 1 2𝑎 × ( 𝑏 √𝑏 2 +4×𝑎×𝑓𝑥 -1)| × ∆𝑏 + | 1 √𝑏 2 +4×𝑎×𝑓𝑥 | × ∆𝑓𝑥
Data for CPDs and 64PPs were fitted by a linear regression 𝑓 = 𝑎 × 𝐷. In this case (9) 𝐷𝑥 = 𝑓𝑥 ̅̅̅̅ 𝑎 and the error is

(10) ∆𝐷𝑥 = | 𝑓𝑥 𝑎 2 | × ∆𝑎 + | 1 𝑎 | ∆𝑓𝑥

Kinetic analysis

Another strategy for the determination of Dx involved a precise kinetic analysis. Because the level of damage was low, we considered that the DNA amount remained constant (DNA0) and set to DNA0=10 6 bases.

CPD

The formation of CPD in DNA was described as a single first order kinetic reaction.

(

) 𝐶𝑃𝐷 = 𝐷𝑁𝐴 0 × (1 -𝑒 -𝑘×𝐷 ) 11
For the determination of a dose Dx leading to a specific level of CPD (fx):

(12) 𝐷𝑥 = -1 𝑘 × 𝑙𝑛 (1 - 𝑓𝑥 𝐷𝑁𝐴 0 )
The error on this value is thus:

(13) ∆𝐷𝑥 = | 1 𝑘×(1-𝑓𝑥) | × ∆𝑓𝑥 + | ln (1- 𝑓𝑥 𝐷𝑁𝐴 0 ) 𝑘 2
| × ∆𝑘 where 𝑓𝑥 is the mean value of the level of CPD in a co-exposure experiment and k the estimated error on the rate k.

64PP and Dewar

Formation of 64PP and Dewar results from two consecutive first order reactions DNA64PPDewar with rate constants k1 and k2 respectively [START_REF] Tuckerman | Chemical Kinetics -More complex reactions[END_REF].

(14) 64𝑃𝑃 = 𝐷𝑁𝐴 0 × 𝑘1 × 𝑒 -𝑘1×𝐷 -𝑒 -𝑘2×𝐷 𝑘2-𝑘1 (15) 𝐷𝑒𝑤𝑎𝑟 = 𝐷𝑁𝐴 0 × [1-(𝑘2×𝑒 -𝑘1×𝐷 -𝑘1×𝑒 -𝑘2×𝐷 )] 𝑘2-𝑘1
For the calculation of k1, we considered that all 64PPs produced at a given dose are either in their initial 64PP forms or as Dewars. Therefore, the sum of the levels of 64PP and Dewar can be described, like CPD, by a first order equation with a reaction constant k1.

(16) (64𝑃𝑃 + 𝐷𝑒𝑤𝑎𝑟) = 𝐷𝑁𝐴 0 × (1 -𝑒 -𝑘1𝐷 )
Log conversion and linear regression yielded the value of k1.

For k2, we converted the exponentials in equation ( 15) by a second order limited development:

(17) 𝐷𝑒𝑤𝑎𝑟 = 𝐷𝑁𝐴 0 × 𝑘1×𝑘2×𝐷 2 2
Linear regression of the data with respect to D 2 provided a fair estimation of k2.

The dose Dx required for a specific level of Dewar (fx) can be determined from equation ( 18):

(18) 𝐷𝑥 = √ 2×𝑓𝑥 ̅̅̅̅ 𝐷𝑁𝐴 0 ×𝑘1×𝑘2
The error on this value is thus:

(19) ∆𝐷𝑥 = | 1 𝑘1×𝑘2 |×∆𝑓𝑥+| -𝑓𝑥 ̅̅̅̅ 𝑘2×𝑘1 2 |×∆𝑘1+| -𝑓𝑥 ̅̅̅̅ 𝑘2 2 ×𝑘1 |×∆𝑘2 𝐷𝑁𝐴 0 ×√ 2×𝑓𝑥 ̅̅̅̅ 𝑘1×𝑘2×𝐷𝑁𝐴 0
For the determination of Dx for 64PPs, we first applied a second order limited development of the exponential functions in equation [START_REF] Tonolli | Lipofuscin generated by UVA turns keratinocytes photosensitive to visible light[END_REF], and obtained equation ( 20):

(20) 64𝑃𝑃 = 𝐷𝑁𝐴 0 × 𝑘1 × 𝐷 -𝐷𝑁𝐴 0 × 𝑘1× (𝑘1+𝑘2) 2 × 𝐷 2
The dose Dx necessary for the induction of a defined level of 64PP (fx) is thus the solution (equation 21) of a second order equation that can be described as the ratio of 2 functions A and B:

(21) 𝐷𝑥 = (-𝑘1×𝐷𝑁𝐴 0 +√((𝑘1×𝐷𝑁𝐴 0 ) 2 -2×𝑓𝑥 ̅̅̅̅ ×(𝑘1 2 +𝑘1×𝑘2)×𝐷𝑁𝐴 0 )) -(𝑘1 2 +𝑘1×𝑘2)×𝐷𝑁𝐴 0 = 𝐴 𝐵
The other solution of [START_REF] Escher | From the exposome to mechanistic understanding of chemical-induced adverse effects[END_REF],

(-𝑘1×𝐷𝑁𝐴 0 -√((𝑘1×𝐷𝑁𝐴 0 ) 2 -2×𝑓𝑥 ̅̅̅̅ ×(𝑘1 2 +𝑘1×𝑘2)×𝐷𝑁𝐴 0 )) -(𝑘1 2 +𝑘1×𝑘2)×𝐷𝑁𝐴 0
, provided aberrant values (maximum Dx for fx=0 and then decreasing Dx for increasing fx).

Calculation of errors requires the derivation of Dx with respect to the k1, k2 and fx.

Consequently, we calculated dA and dB for each of these 3 parameters:

(22') 𝑑𝐴(𝑘1) = -𝐷𝑁𝐴 0 + 𝑘&×𝐷𝑁𝑂 𝑜 é +𝑓𝑥×𝐷𝑁𝐴 0 ×(2 𝑘1+𝑘2) √𝑘1 2 ×𝐷𝑁𝐴 0 2 -2×𝑓𝑥×(𝑘1 2 +𝑘1×𝑘2)×𝐷𝑁𝐴 0 (22'') 𝑑𝐴(𝑘2) = -𝑓𝑥×𝐷𝑁𝐴 0 ×𝑘1 √𝑘1 2 ×𝐷𝑁𝐴 0 2 -2×𝑓𝑥×(𝑘1 2 +𝑘1×𝑘2)×𝐷𝑁𝐴 0 (22''') 𝑑𝐴(𝑓𝑥) = -(𝑘1 2 +𝑘1×𝑘2)×𝐷𝑁𝐴 0 √𝑘1 2 ×𝐷𝑁𝐴 0 2 -2×𝑓𝑥×(𝑘1 2 +𝑘1×𝑘2)×𝐷𝑁𝐴 0 (23') 𝑑𝐵(𝑘1) = -(2 𝑘1 + 𝑘2) × 𝐷𝑁𝐴 0 (23'') 𝑑𝐵(𝑘2) = -𝑘1 × 𝐷𝑁𝐴 0
Combination of equations (22') to (23'') provided the error on Dx:

(24) ∆𝐷𝑥 = |𝐷𝑥 × ( 𝑑𝐴(𝑘1) 𝐴 - 𝑑𝐵(𝑘1) 𝐵 )| × ∆𝑘1 + |𝐷𝑥 × ( 𝑑𝐴(𝑘2) 𝐴 - 𝑑𝐵(𝑘2) 𝐵 )| × ∆𝑘2 + |𝐷𝑥 × 𝑑𝐴(𝑓𝑥) 𝐴 | × ∆𝑓𝑥

Combination index

The Combination Index CI was calculated for each experiment involving combination of UVB and UVA using equation ( 25 3), ( 7), ( 9), ( 12), [START_REF] Douki | Dewar valence isomers, the third type of environmentally relevant DNA photoproducts induced by solar radiation[END_REF] or [START_REF] Lasch | Mixture effects of chemicals: The difficulty to choose appropriate mathematical models for appropriate conclusions[END_REF] depending on the data modeling used and the photoproduct considered.

Using the estimated errors Dx calculated for UVB and UVA alone (equations ( 5), ( 8), ( 10), ( 13), [START_REF] Taylor | DNA, light, and Dewar pyrimidones: the structure and biological significance of TpT3[END_REF] or ( 24)), and the dose D of UVB and UVA used in the combined exposure, the error on CI was: These fits of the data were then used to calculate the CI for the different doses of combined UVB+UVA radiation. We were faced with lack of formation of 64PPs and Dewars in the samples exposed to UVA only. This situation is not a priori compatible with the calculation of CI where the ratio 𝐷 𝐷𝑥 is taken into account for both factors. To overcome this limitation, we reasoned that a lack of formation could be seen as the necessity to apply an infinite dose to induce an effect. Consequently, the Dx value in the ratio involved in the calculation of CI is infinite and the ratio is null. We thus did not consider the term

𝐷 𝑈𝑉𝐴 𝐷 𝑈𝑉𝐴𝑥
in the calculation of CI for 64PPs, Dewars and CC CPD. Another improvement of the CI formalism in the present study was the accurate estimation of the error of the value of CI. A possible approach could be a replication of the whole experiment and the calculation of the mean and the variance. However, such an approach is not satisfactory because it does not really take into account the error on the MEE parameters and mostly reflects the variation in the results of the combination experiment.

Therefore, we rather took into account the propagation of the estimated errors of the fitted parameters of the linear regression aimed at calculating the m and Dm coefficients of the MEE.

A first observation regarding the calculated CI in this experiment was that the values were similar at all doses and close to 1 for the four studied CPDs (Fig. 2). This reflects that, in the UVB and UVA ranges applied, formation of CPDs is a simple first order reaction. The photoreversion observed with UVC radiation [START_REF] Johns | Photochemistry of thymine dimers[END_REF][START_REF] Lemaire | Kinetic analysis of the deamination reactions of cyclobutane dimers of thymidylyl-3',5'-2'-deoxycytidine and 2'-deoxycytidine-3,5'thymidine[END_REF], which is explained by a residual absorption by CPD of these high-energy photons, does not take place. The present results also confirmed that UVA induces the formation of T-containing CPDs but in much lower yield than UVB. The observed ratio here was approximately 2000 for TT CPD. Consequently, additivity is observed for CPD in case of co-exposure to UVB and UVA and the CI value is 1. In contrast to the formation of CPDs, the formation of 64PPs and Dewars was greatly affected by the presence of UVA during UVB irradiation. 64PPs are efficiently produced by UVB but not UVA. Both UVB and UVA are absorbed by the pyrimidone ring of 64PPs and convert them into Dewars. In isolated 64PPs like photoproducts of dinucleoside monophosphates, this reaction exhibits a maximal efficiency in the UVB range as 64PPs have a maximal absorption at approximately 320 nm [START_REF] Douki | Characterization of the (6-4) photoproduct of 2'-deoxycytidylyl-(3'-5')-thymidine and of its Dewar valence isomer[END_REF][START_REF] Taylor | DNA, light, and Dewar pyrimidones: the structure and biological significance of TpT3[END_REF][START_REF] Taylor | Quantitative conversion of the (6-4) photoproduct of TpdC to its Dewar valence isomer upon exposure to simulated sunlight[END_REF][START_REF] Lemaire | Quantum yields and secondary photoreactions of the photoproducts of dTpdT, dTpdC and dTpdU[END_REF]. However, UVA is more efficient than UVB at photoisomerizing 64PPs in DNA because of the shielding effects of the overwhelming amount of the normal bases in the latter wavelengths range. This is observed in isolated as well as in cellular DNA [START_REF] Douki | Relative contributions of UVB and UVA to the photoconversion of (6-4) photoproducts into their Dewar valence isomers[END_REF] and could be modeled based on available absorption features and quantum yield data [START_REF] Douki | Dewar valence isomers, the third type of environmentally relevant DNA photoproducts induced by solar radiation[END_REF]. The calculated CIs for the TT and TC 64PPs and Dewars unambiguously reflect this chain of reactions (Table 1). The value of the CI is below 1 for Dewars. The mean value was 0.3 and 0.4 for TT Dewar and TC Dewar, respectively, and thus corresponds to a synergy effect in the combined exposure, namely a favored formation in the presence of UVA. The opposite trend is observed for 64PPs since their CI are larger than 1. The CI analysis even reflects known differences between the photochemistry of TC and TT 64PPs. The antagonism of the formation of 64PP by UVA and the synergistic formation of Dewar are stronger for the TT than the TC photoproducts. Actually, it was observed, from isolated DNA (44) to mammalian cells [START_REF] Clingen | Induction of cyclobutane pyrimidine dimers, pyrimidine(6-4)pyrimidone photoproducts, and Dewar valence isomers by natural sunlight in normal human mononuclear cells[END_REF][START_REF] Qin | Detection of ultraviolet photoproducts in mouse skin exposed to natural sunlight[END_REF] and marine microorganisms collected on the field [START_REF] Meador | The significance of the Dewar valence photoisomer as a UV radiation-induced DNA photoproduct in marine microbial communities[END_REF], that the yield of conversion into Dewars was larger for TT 64PP than TC 64PP. The quantum yield for the two photoisomerization reactions are similar with values of 2.0 10 -4 and 1.8 10 -4 , respectively [START_REF] Lemaire | Quantum yields and secondary photoreactions of the photoproducts of dTpdT, dTpdC and dTpdU[END_REF]. The absorption coefficient of TC 64PP is 3 times lower than that of TT 64PP [START_REF] Douki | Far-UV-induced dimeric photoproducts in short oligonucleotides: Sequence effects[END_REF], which partly explains the difference in the yield of photoisomerization. A maximum absorption is observed at 325 nm for TT 64PP [START_REF] Taylor | DNA, light, and Dewar pyrimidones: the structure and biological significance of TpT3[END_REF] and 315 nm (32) for TC 64PP. Therefore, the lower absorption coefficient of TC 64PP is compensated by the emission of the UVB source used in this work that exhibits a maximum at 312 nm. More interesting for the topic of our work is that calculations based on known absorption coefficients and quantum yields provided evidence that the photoisomerization yield of TC 64PP is still 3-fold lower than that of TT 64PP in combination of 300 and either 315, 320 or 340 nm light [START_REF] Douki | Relative contributions of UVB and UVA to the photoconversion of (6-4) photoproducts into their Dewar valence isomers[END_REF]. This larger synergy in the isomerization of TT 64PP compared to TC 66PP is reflected by the lower value of CI for TT Dewar than TC Dewar. Another observation is the increase in the value of the CI of 64PPs when the applied dose increased. This feature may be explained by the fact that, upon exposure to combination of UVB and UVA, the level of 64PPs tends to a plateau while the increase is more monotonous with pure UVB. Consequently, the antagonist effect is more pronounced when the dose increases. Regarding the error on the determination of the CI, a general trend observed for all photoproducts is that it is larger at the lowest doses. This can be explained by a less accurate determination of the amount of photoproducts or a less reproducible applied UV dose. In addition, the weight of the low values at the lowest dose is lower than that at high dose in the regression providing the parameters of the MEE. It is thus expected that the accuracy is lower at low dose. The results clearly show that calculation of the CI based on the MEE as defined by Chou [START_REF] Chou | Theoretical basis, experimental design, and computerized simulation of synergism and antagonism in drug combination studies[END_REF] is well suited for establishing additive or non-additive effects in photochemically-induced processes like formation of DNA damage. CI nicely reflects the additivity between UVB and UVA in the induction of CPDs, the lower levels of 64PPs in the presence of UVA and the larger amounts of Dewars upon co-exposure to UVB and UVA. The present work illustrates the ease of the procedure that requires only three experiments with a few doses to obtain an indication on the occurrence of synergistic or antagonist effects over a wide range of doses. In the present work, we could explore a dose range corresponding to a factor of 45 between the lowest and the largest dose with only 6 points for UVB, UVA and UVB+UVA.

Calculation of CI with other fitting functions

The results presented above for UVB and UVA is an illustration, in addition to countless examples with drugs and chemical, of the power of the CI approach to identify synergism and antagonism in mixtures. The definition of this parameter was inferred from extensive theoretical work based on the mass action law and the concept of the MEE. Yet, the MEE was designed for drug-target interactions and cannot reflect all possible biological responses. This is the case for all the cellular processes that do not exhibit a dose-response shape with a plateau and that cannot be normalized between 0 and 1. An important example is that of gene expression that can be larger than one when increased with respect to a control sample and that do not have a theoretical upper limit. We reasoned that in these cases, the CI can still be calculated because it only depends on ratio of doses. We thus explored whether the role of the MEE in the initial strategy developed by Chou ( 25) could be played by other mathematical functions. This would allow fitting any shape of dose-dependent cellular responses. In addition, the absolute requirement of responses ranging between 0 and 1 permitting the MEE to be applied can be avoided by using other equations. The limiting ingredient to the determination of CI is that the fitted function for the effect with respect to the dose may easily be inverted to determine Dx for the observed effect fx of mixtures. It is yet important to limit the number of fitting parameters in order to avoid overfitting the experimental data of the pure dose effects.

We thus used the set of experimental results analyzed above with the MEE equation with other functions aimed at fitting the data for pure UVA and UVB and calculating Dx from fx. We first applied polynomial regressions. Linear regressions were used for CPDs and 64PPs, and second order polynomial regressions for Dewars. In a first attempt, second order polynomial regressions were also used for 64PPs but errors were very large and the obtained CIs close to those obtained with linear regression (data not shown). The CI values obtained for the four CPDs, and for TT and TC 64PPs and Dewars were very similar whether a polynomic regression (Table 2) or the MEE were used (Table 1). The mean ratio between values provided by the MEE and the polynomial regression is 1.06. This result shows that the CI can be used with any fitting function, even if it is not based on known physical-chemical processes like MEE that is based on the mass action law. In addition, the values of the level of photoproducts used in the polynomial regressions were expressed as number of photoproducts per million bases and were thus larger than 1 (up to 9000 per 10 6 bases for TT CPD). Because the photochemical reactions leading to the formation of the three types of photoproducts are well known, we could also apply a more rigorous approach for the fit of the experimental data by using chemical kinetic equations. Formation of CPDs was described as a simple first order reaction characterized by a reaction constant k. In contrast, formation of 64PPs and Dewars were described as two consecutive reactions with reaction constants k1 and k2. The constant k1 was related to the initial formation of 64PPs and second k2 reflected their conversion into Dewar. As described in the "Data modeling" part, these equations were used to fit the data and calculate the Dx values necessary for the determination of the CI. Like for polynomial regression, the obtained CI values (Table 2) were very close to those obtained using the MEE (Table 1). The mean ratio between values provided by the MEE and the kinetic analysis is 1.07. Obviously, this type of kinetic data analysis is only possible when the basic steps of a mechanisms are well identified. In this case, the use of a strategy like the CI is thus not necessary to predict occurrence of non-additivity. However, the fact that similar results

were obtained for the CI with various fitting strategies like MEE or polynomial regressions on the one hand, and more realistic calculations based on kinetic parameters on the other, validates the two former approaches. Interestingly, the kinetic analysis emphasized a limitation to the CI determination that concerns all fitting functions. The dose dependent-formation of 64PPs exhibits a bell-shaped curve. Calculation of the theoretical level of TT 64PP for large doses based on equation [START_REF] Dreaden | N-formylkynurenine as a marker of high light stress in photosynthesis[END_REF] shows that a maximal level is reached at approximately 200 kJ m -2 . At this dose, the level of photoproduct is 1180 TT 64PP per million bases. Consequently, above this dose, the calculation of Dx will provide 2 relevant values for a mixture effect fx and it will not be possible to calculate a CI. This was not the case in our experiments where the dose was lower than 12 kJ m -2 UVB. This example of the use of the kinetic equations illustrates that the use of functions other than the MEE for the determination of CI has to be carried out at doses below maximal responses for bell-shaped curves.

Interesting observations can also be made regarding the errors on the CI. When the latter parameter was calculated on the basis of the MEE, the lowest relative errors were obtained for Dewars, with values of approximately 30% at the largest dose. However, for both polynomial regression and kinetic analysis, this error is on the average 5 times lower than with the MEE. A similar trend was observed for 64PPs when a linear regression was used for the determination of the CI compared to MEE. The relative errors on CI are 12% and 48%, respectively, for TT 64PP at the largest dose. The corresponding values are 19 and 50 % for TC 64PP. Comparison of errors was also in favor of the kinetic analysis versus MEE, although to a lesser extent than the polynomial regressions. With the kinetic analysis, the relative errors on the CI for TT 64PP

and TC 64PP at the largest dose were 25 and 27 %, respectively. From the back propagation equations, it is possible to estimate the relative impact of the experimental data and the fitting parameters on the error on Dx (Dx) and consequently on the CI error (CI). If Dx is dominated by the parameters of the fits, other fitting functions may require to be considered or experiments of pure factors with a larger dose range. On the contrary, if Dx is dominated by experimental data on mixture effects, new experiments of mixtures with more replicates may be required. For TT 64PP at the largest dose, Dx determined with the MEE originates in 42%, 32% and 26% from error on Dm, m and fx, respectively. In this case, improvement of the accuracy of the fit seems necessary. For the kinetic analysis approach applied to TT 64PP, the proportions of Dx originating from k1, k2 and fx were 22%, 31% and 47% at the largest dose.

The result show a strong contribution of the fit. For the second order polynomial fit of TT 64PP

(not discussed in this work), CI was 2000 for a CI of 4.5 and the error on the quadratic coefficient represented 99.9% of Dx. With the linear regression, a much lower error was obtained (0.57). The contribution of the errors to Dx were 14 and 86 % for the value of the slope and fx, respectively. In this case, increasing the quality of the experimental results of mixtures, for example by using more replicates, could improve the estimation of CI. The second order polynomial fit was found to be very accurate for TT Dewar with an error of 4% on CI at the largest dose. In this case the contribution of the errors to Dx were 40%, 56% and 4% for quadratic term, the linear term and fx, respectively. These observations point again to a significant contribution of the quality of the fit to the error on the CI. Altogether, these results suggests that Dx and CI are mostly explained by errors on the fit, especially when the overall errors are large. This shows that the choice of the fitting function is a key parameter in the accuracy of the determination of the CI. In the average, the MEE predicts CI with an error on the average 4 times larger than polynomial regressions or kinetic analysis, while the CI values are similar with all approaches.

Dependence of CI on the irradiation conditions

We then performed a series of experiments still involving a combination of UVA and UVB in order to determine whether the CI was sensitive to the exposure protocols. In this experiment and in all the rest of the work, we relied on the initial strategy designed by Chou et al. [START_REF] Chou | Theoretical basis, experimental design, and computerized simulation of synergism and antagonism in drug combination studies[END_REF][START_REF] Chou | The mass-action law based algorithm for cost-effective approach for cancer drug discovery and development[END_REF] for the calculation of the CI with the MEE. We first investigated whether the CI varied according to the UVA/UVB ratio. We then studied the impact of the order in which UVA was combined with UVB, either simultaneously or one after the other. For these experiments, only one sample was collected at each dose. Consequently, we did not calculate errors.

Evolution of the CI at different UVA to UVB ratios

In the previous study, UVA and UVB were applied at a constant ratio of 18:1. Yet, it is anticipated that the efficiency of conversion of 64PPs into Dewars depends on the relative intensities of UVB and UVA. Therefore, we calculated the CI through a MEE fitting for different exposure conditions exhibiting increasing UVA to UVB ratios. For this purpose, we kept the UVA fluence constant and decreased that of the UVB source. The results unambiguously showed a decrease of the value of the CI related to Dewars with increasing ratio, namely an increase in the synergistic effect, when the UVA to UVB ratio increased (Fig. 3). Conversely, the CI for TT and TC 64PPs was larger than 1 under all exposure conditions, showing the antagonist effect of UVA. The effect was not as strong as those shown on Figure 1 because the total UV doses applied were lower. Like in our experiments at constant UVA/UVB ratio, the values of the CI for TC Dewar were larger than those for the TT Dewar.

The mean ratio between the CI of TC Dewar and TT Dewar at different applied doses ranged from 1.2 for a UVA/UVB ratio of 3.3, to 1.9 for a UVA/UVB ratio of 21.8. These observations reflect the more efficient isomerization of TT 64PP in the UVA range compared to TC 64PP.

This experiment illustrates that the CI is more than a semi-quantitative indication of the non-additivity of response in co-exposures. The modulation of DNA photochemistry induced by changing the ratio between the fluence of UVB and UVA by a value of 2 or 3 was enough to be reflected in the values of the CI.

Effect of the order of exposure on the value of CI

In the subsequent study, we compared the results of UVB and UVA co-exposure experiments where the two radiation ranges were applied either simultaneously or one after the other.

Experiments involving pure UVA and UVB were also performed in order to calculate the CI.

As observed in all the other experiments, the CI values determined for CPDs were close to 1, irrespectively of the irradiation conditions and of the dose (data not shown). Interestingly, the total dose of UVA and UVB received by the DNA samples was the same in the UVAUVB, UVB+UVA and UVBUVA protocols. Yet, differences were observed in the photoisomerization yield. In the first case (UVAUVB), the efficiency is not different than with UVB alone as shown by the CI close to 1 for 64PPs and Dewars. This is expected since UVA is applied first and cannot be absorbed by 64PPs produced by UVB. This was not the case for the two other irradiation protocols. For TT 64PP, the value of the CI increased continuously from 0.98 to 1.30 with increasing dose for UVB+UVA and from 1.26 to 1.69 for UVBUVA.

The same trend was observed for TC 64PP. The CI values obtained at the lowest and largest UV doses were 1.00 to 1.14 with increasing dose for UVB+UVA and from 1.04 to 1.29 for UVBUVA. A mirror effect was observed on the Dewar data. As shown on figure 4, the CI value for both TT and TC Dewars were 1 on the average in the UVAUVB protocol, but lower in the UVB+UVA and UVBUVA experiments. The mean values are 0.44 and 0.28 for TT Dewar for UVB+UVA and UVBUVA, respectively. The corresponding values for TC Dewar are 0.74 and 0.56. (Fig. 4). In summary, the CI for 64PPs was larger for UVBUVA than UVB+UVA while the opposite trend was observed for Dewars. Interestingly, the CI of the TT 64PP is more dose-dependent than that of TC 64PP. This can be explained by the fact that the photoisomerization reaction is more efficient. Consequently, the trend to a plateau for 64PP and the quadratic shape of the formation of Dewar are more pronounced for TT than TC. In the calculation of the CI, this is reflected by a larger dose dependence for TT than TC photoproducts. The bulk of these observations unambiguously shows that the photoisomerization is more efficient with UVBUVA than UVB+UVA. This result can be explained by the fact that when UVA is applied after UVB, all UVB-induced 64PPs absorb the maximal amount of UVA. When both wavelength ranges are applied simultaneously, the same amount of 64PPs as in the UVBUVA design are produced. However, those formed close to the end of irradiation are exposed to a lower dose of UVA than those produced at the beginning. Consequently, their yield of photoisomerization is lower. A more comprehensive explanation can be found in figure 5 UVBUVA: UVB alone followed by UVA alone; UVAUVB: UVA alone followed by UVB alone. The graphs were obtained by using equations ( 14) and [START_REF] Dreaden | N-formylkynurenine as a marker of high light stress in photosynthesis[END_REF]. The value of k1 was obtained in the UVB experiment. A value k2UVB was determined for pure UVB as explained in the calculation part. A similar approach was applied for the determination of k2UVB+UVA in the UVB+UVA experiment. The difference between these 2 values yielded kUVA used for the fit of the UVBUVA data. The values of k1, k2UVB, k2UVB+UVA and k2UVA were found to be 4 10 -7 , 0.002, 0.010 and 0.008 min -1 , respectively.

Combination of 3 UV sources

The results presented above involved only a UVB and a UVA source. Yet, photobiological processes could be modulated by combinations of more than 2 ranges of wavelengths. We took advantage that the CI can be used with more than two factors to explore a ternary combination of UV radiation. Using our exposure device fitted with specific filters, we exposed DNA to UVB (310±5 nm), UVA2 (340±13 nm) and UVA1 (386±13.5 nm). In addition to each radiation alone and to the ternary combination, we exposed solutions of isolated DNA to a combination of UVB and UVA2 (310+340 nm) and a combination of UVB and UVA1 (310+386 nm). As observed with the full UVA spectrum, neither UVA1 nor UVA2 led to the formation of 64PPs and Dewars. We observed that the CI was 1 for 64PPs and Dewars for the 310+386 nm combination (Fig. 6), suggesting that UVA1 is not efficient in the photoisomerization of 64PP.

In contrast, the CI for Dewars in the 310+340 nm experiments were below 0.5 and statistically significantly lower than those observed for 310+386 nm. The same observations were made in the ternary combination, showing that addition of UVA1 to the UVB/UVA2 radiation did not increase the photoisomerization yield of 64PPs. It should be noticed that a statistically significant larger CI was obtained for TC 64PP in the 310+340 nm irradiation compared to 310+386, as well as between the 310+386 and the 310+340+386 experiments. The latter difference was also statistically significant for TT 64PP, and the comparison between 310+340 and 310+386 exhibited the same trend of a CI closer to 1 under the latter condition. These results strongly suggest an antagonist effect of UVA2, but not of UVA1, on the formation of 64PPs. Combined with observation of synergistic effects of UVA2 but not UVA1 on the formation of Dewars, the CI analysis allowed us to conclude that UVA1 does not participate to the photoisomerization of 64PPs, in agreement with their known absorption properties.

CONCLUSION

In the present work, we proposed the use of the combination index as a tool to identify non additive effects upon co-exposure to different radiation ranges. Application of this approach to the well-known photochemistry of DNA shows that the CI strategy is as efficient with light as it is with chemicals. We also provided some improvements to the original design by showing that fitting functions other than the MEE can be used. In particular, we show that regressions that are not necessarily based on physico-chemical principles like the mass action law for MEE or chemical kinetics analysis are well suited. In our case, linear and second order polynomial fits of the data provided CI values similar to those obtained with the MEE or kinetic analysis, with the lowest incertitude. In addition, these fitting functions can exhibit any dose-response curves, except bell-shaped responses. Another advantage of the use of such fitting functions is that the value of the considered effect does not have to be lower than 1 as in the MEE, which would be a strong limitation for example for the induction of all photobiological effects normalized to a control like gene expression or protein synthesis. Further developments will involve application of the combination index strategy to investigations of the effect of complex exposures on living systems with biologically relevant endpoints. Another long term project would be to incorporate the non-additivity brought by CI methodology in a generalized action spectra for the estimation of dose effects with complex light sources like solar spectra.

. 2 = 1 . 2 1 :

 2121 Scheme 1: Principle of the determination of the combination index CI. (a) The amplitude of the studied effect fa is determined for factors 1 and 2 applied alone on a range of concentrations that can be different for the two factors. (b) Experiments are performed with mixtures of factors 1 and 2 exhibiting a constant ratio D1/D2. The CI can be determined for each of the mixtures. Those have a concertation D1+D2 and induce an effect fx. (c) Data from (a) are fitted by the MEE. The latter equation provides the values of the concentration of pure factors 1 (Dx1) and 2 (Dx2) leading to the effect fx. (d) The CI is calculated and the conclusion on additivity, synergy or antagonism is deduced.

  with DUVB and DUVA representing the doses of UVB and UVA radiation applied to the sample in the combination experiments, and DUVBx and DUVAx the doses of pure UVB and UVA inducing the same effect fx as the combination (DUVB+DUVA). DUVBx and DUVAx were calculated using either equation (

( 26 ) 2 )

 262 ∆𝐶𝐼 = (𝐷 𝑈𝑉𝐵 × ∆𝐷𝑈𝑉𝐵𝑥 𝐷𝑈𝑉𝐵𝑥 2 ) + (𝐷 𝑈𝑉𝐴 × ∆𝐷𝑈𝑉𝐴𝑥 𝐷𝑈𝑉𝐴𝑥 RESULTS AND DISCUSSION Effects of UVB and UVA co-exposure shown by the Combination Index and the Median Effect Equation.In a first experiment, we investigated the effect of a combined exposure to UVB and UVA through the classical CI approach with a modeling of the pure UVB and UVA data with the MEE. DNA samples in aqueous solution were exposed to increasing doses of UVB (0.3-12 kJ m -2 ) or UVA (5-212 kJ m -2 ). In a third experiment, DNA was exposed simultaneously to the same increasing doses of UVB and UVA as those applied alone. The overall UV dose ranged thus between 5.3 and 223 kJ m -2 . The parameters of the MEE, which are necessary to the determination of the Dx values used in the calculation of the CI value, were calculated for UVB for 8 photoproducts: TT CPD, TT 64PP, TT Dewar, TC CPD, TC 64PP, TC Dewar, CT CPD and CC CPD. As previously reported, CC CPD, 64PPs and Dewars were not detected in the samples exposed to UVA with our specific and sensitive HPLC-MS/MS assay[START_REF] Courdavault | Larger yield of cyclobutane dimers than 8 oxo-7,8-dihydroguanine in the DNA of UVAirradiated human skin cells[END_REF][START_REF] Mouret | Cyclobutane pyrimidine dimers are predominant DNA lesions in whole human skin exposed to UVA radiation[END_REF][START_REF] Mouret | UVA-induced cyclobutane pyrimidine dimers in DNA: a direct photochemical mechanism?[END_REF].Therefore, the MEE was determined for UVA only for TT CPD, TC CPD and CT CPD. The fit of UVB data are shown on figure1. The MEE slightly deviated from the experimental data for CPDs at the largest UVB dose but nicely reproduced the shape of the dose curve formation of 64PPs and Dewars. A larger scattering was observed for UVA (data not shown) but the impact of these values on the final CI is very low.

Figure 1 :

 1 Figure 1: Fit of the dose-dependent formation of pyrimidine dimers in DNA exposed to UVB alone by the median effect equation (MEE) for a) TT and TC CPDs, b) TT 64PP and Dewar c) TC 64PP and Dewar. The marks represent the experimental data while the plain lines are the MEE fits.

Figure 2 :

 2 Figure 2: CI calculated for CPD upon exposure of DNA to UVB and UVA in a 1:18 ratio. The reported values are the means (± standard deviation) of results obtained in duplicate at six different UV doses.

Figure 3 :

 3 Figure 3: Value of the CI for a) TT Dewar and b) TC Dewar in DNA samples exposed to radiation with various UVA/UVB ratios.

Figure 4 :

 4 Figure 4: variation of CI of TT and TC Dewars for different irradiation protocols. UVB and UVA were applied simultaneously (UVB+UVA), or successively UVB UVA or UVAUVB. Dashed lines are linear fits to illustrate the trends of CI as function of the total UV dose.

  figure 4 were the final values for each irradiation time (5+5, 10+10, 20+20 or 40+40 min).

Figure 5 :

 5 Figure 5: Formation of a) TT 64PP and b) TT Dewar in isolated DNA exposed to either UVB alone or to UVB and UVA in different combinations. UVB+UVA: simultaneous exposure;

Figure 6 :

 6 Figure 6: CI determined for TT and TC 64PPs and Dewars in DNA samples exposed to combination of UVB, UVB and UVA1 and UVB and UVA2. The reported results are means ± SEM determined in triplicate at 4 different total UV doses (n=12). The statistical significance of the differences, evaluated by a one-way ANOVA, were p<0.05 (*) or p<0.01 (**).

Table 1 :

 1 Values of the combination index (CI) determined for increasing doses of UVA andUVB applied in a 18:1 ratio. The reported total UV dose is expressed in kJ m -2 . The errors were obtained as described in the calculation section and included errors on the parameters of the MEE and the Dx dose.

			CI calculated by the Median Effect Equation
	UV dose	TT 64PP	TT Dewar	TC 64PP	TC Dewar	TT CPD	TC CPD
	5	1.12±0.63	0.32±0.11	1.23±0.75	0.44±0.20	0.83±0.12	0.83±0.23
	12	1.27±0.57	0.31±0.12	1.28±0.67	0.48±0.22	0.99±0.18	0.87±0.23
	37	1.53±0.71	0.29±0.08	1.33±0.62	0.45±0.19	1.13±0.21	1.05±0.29
	75	2.04±0.96	0.28±0.08	1.30±0.73	0.38±0.16	1.07±0.15	1.03±0.26
	149	3.30±1.65	0.30±0.08	1.44±0.74	0.34±0.13	1.03±0.16	1.09±0.44
	223	4.75±2.34	0.33±0.08	1.69±0.82	0.32±0.10	1.04±0.19	1.07±0.30

Table 2 :

 2 Values of the combination index (CI) determined for increasing doses of UVA andUVB applied in an 18:1 ratio, with the use of data fitting other than the MEE. The reported total UV dose is expressed in kJ m -2 . The errors were obtained as described in the calculation section and included errors on the parameters of the polynomial regression or kinetic equations, of the Dx dose and on the final calculation of CI.

		CI calculated by linear or second order polynomial regression
	UV dose	TT 64PP	TT Dewar	TC 64PP	TC Dewar	TT CPD	TC CPD
	5	0.85±0.14	0.28±0.04	0.96±0.24	0.41±0.27	0.92±0.14	0.71±0.20
	12	1.01±0.06	0.27±0.03	1.05±0.18	0.41±0.17	1.06±0.15	0.76±0.12
	37	1.29±0.13	0.27±0.01	1.17±0.15	0.40±0.08	1.16±0.10	0.94±0.06
	75	1.75±0.20	0.30±0.02	1.19±0.28	0.38±0.06	1.06±0.18	0.95±0.02
	149	2.87±0.42	0.35±0.02	1.37±0.28	0.39±0.04	1.00±0.13	1.02±0.03
	224	4.15±0.57	0.40±0.01	1.64±0.30	0.40±0.02	0.99±0.06	1.02±0.13
			CI calculated by kinetic analysis
	UV dose	TT 64PP	TT Dewar	TC 64PP	TC Dewar	TT CPD	TC CPD
	5	1.07±0.30	0.21±0.01	1.07±0.35	0.25±0.01	0.92±0.14	0.71±0.21
	12	1.27±0.23	0.24±0.02	1.17±0.29	0.32±0.02	1.06±0.15	0.76±0.12
	37	1.60±0.35	0.27±0.01	1.29±0.27	0.38±0.02	1.16±0.11	0.94±0.06
	75	2.15±0.51	0.30±0.02	1.30±0.41	0.38±0.03	1.07±0.18	0.95±0.03
	149	3.49±0.95	0.35±0.02	1.46±0.43	0.40±0.02	1.00±0.13	1.02±0.03
	224	5.03±1.33	0.41±0.02	1.71±0.47	0.42±0.01	0.99±0.06	1.02±0.13

ACKNOWLEDGMENTS

This work was supported by the Labex ARCANE and the CBH-EUR-GS (ANR-17-EURE-0003).