Youqiang Yao

Pengcheng Shi

Mingda Chen

Gang Chen

Cong Gao

Philippe Boisse

Yingdan Zhu
email: y.zhu@nimte.ac.cn

Experimental and numerical study on Mode I and Mode II interfacial fracture toughness of co-cured steel-CFRP hybrid composites

Interfacial fracture toughness of co-cured steel-carbon fiber reinforced plastic (CFRP) hybrid composites were investigated in this paper. To illustrate the effect of the interlayer on the fracture toughness, steel-CFRP hybrid composites were prepared by different manufacturing processes based on steel surface treatment (abrasion or grit blasting) and adhesive-bonding process. The experimental results of double cantilever beam (DCB) tests and end notched flexure (ENF) tests demonstrate that, the Mode I and Mode II interfacial fracture toughness of the hybrid composites can be improved by using a grit blasting surface treatment on steel and introducing an adhesive layer at the steel/CFRP interface. The hybrid composites mainly show fiber/epoxy interfacial failure of CFRP under Mode I loading conditions, while it mostly exhibits adhesive failure of steel/CFRP interface under Mode II loading condition. Moreover, the interfacial tensile strengths of steel-CFRP hybrid composites are predicted by finite element analysis, and both experimental and numerical results confirm the improvement of interfacial fracture toughness.

Introduction

With fiber-metal laminates (FMLs) showing great application prospects in aerospace industries, metal-CFRP hybrid composites have also attracted great scientific and industrial interests in automotive structural applications [START_REF] Sinmazçelik | A review: fibre metal laminates, background, bonding types and applied test methods[END_REF][START_REF] Marsh | Composites and metalsa marriage of convenience?[END_REF]. Combining the outstanding high specific strength and fatigue resistance of fiber reinforced composites with the high bearing strength and impact resistance ductility of metal alloys, metal-CFRP hybrid composites possess lightweight and performance advantages that cannot be achieved by metals or advanced composites individually.

Nevertheless, the manufacturing of metal-CFRP hybrid composites still faces many challenges, such as weak bonding of heterogeneous material, residual stress caused by thermal expansion coefficient mismatch and galvanic corrosion caused by potential difference. The major defect is the relatively low bonding strength between metal substrate and CFRP laminate. Among a number of adhesion mechanisms, adsorption and mechanical interlock are well-known to be the dominant mechanisms that contribute to the adhesion strength of metal-polymer interfaces [START_REF] Kim | Evaluation of mechanical interlock effect on adhesion strength of polymer-metal interfaces using micro-patterned surface topography[END_REF]. Thus, the adhesive optimization and surface treatment of metal substrate are two effective ways to improve bonding performance. On one hand, the adhesives should have good wettability with adherends. Relative to adhesives in a semi-solid state, liquid adhesives with lower surface tension than adherends are more likely to ensure uniform wetting of entire surfaces and improve molecular contacts between adherends, thereby increasing the joint strength [START_REF] Pramanik | Joining of carbon fibre reinforced polymer (CFRP) composites and aluminium alloysa review[END_REF]. On the other hand, to facilitate the bonding between metals and composites, many kinds of surface treatment methods for metal substrate have been developed including mechanical [START_REF] Rudawska | The effect of sandblasting on surface properties for adhesion[END_REF][START_REF] Chander | Effects of grit blasting on surface properties of steel substrates[END_REF][START_REF] Teng | Treatment of steel surfaces for effective adhesive bonding[END_REF][START_REF] Islam | Influence of metal surface preparation on its surface profile, contact angle, surface energy and adhesion with glass fibre prepreg[END_REF][START_REF] Pan | Effect of surface roughness on interlaminar peel and shear strength of CFRP/Mg laminates[END_REF][START_REF] Islam | Effects of hygrothermal and ambient humidity conditioning on shear strength of metal-GFRP single lap joints co-cured in and out of water[END_REF], chemical [START_REF] Dawood | Environmental durability of a CFRP system for strengthening steel structures[END_REF][START_REF] Ebnesajjad | Chapter 7 -surface preparation of metals[END_REF], electrochemical [START_REF] Baldan | Adhesively-bonded joints and repairs in metallic alloys, polymers and composite materials: adhesives, adhesion theories and surface pretreatment[END_REF][START_REF] Teng | Strengthening of steel structures with fiber-reinforced polymer composites[END_REF] treatment and so on. Among these methods, grit blasting is effective to achieve a coarse and high-energy metal surface. It has been widely used since it is non-polluting, easily adaptable, and economically viable [START_REF] Rudawska | The effect of sandblasting on surface properties for adhesion[END_REF][START_REF] Chander | Effects of grit blasting on surface properties of steel substrates[END_REF][START_REF] Teng | Treatment of steel surfaces for effective adhesive bonding[END_REF][START_REF] Islam | Influence of metal surface preparation on its surface profile, contact angle, surface energy and adhesion with glass fibre prepreg[END_REF]. Islam and Tong [START_REF] Islam | Effects of hygrothermal and ambient humidity conditioning on shear strength of metal-GFRP single lap joints co-cured in and out of water[END_REF] indicated that, compared with needle gunning and wire brushed, grit blasting treated mild steel substrate possessed superior shear bond strength with glass epoxy prepreg composites even at different environment.

For metal-CFRP hybrid composites, interfacial fracture or delamination is one of the major failure modes, which may easily occur at the interface of the two dissimilar materials and result in severe degradation of overall structural strength [START_REF] Sinmazçelik | A review: fibre metal laminates, background, bonding types and applied test methods[END_REF][START_REF] Pramanik | Joining of carbon fibre reinforced polymer (CFRP) composites and aluminium alloysa review[END_REF]. In view of fracture mechanics, interfacial fracture toughness is an important parameter for measuring crack propagation resistance of the laminated composites [START_REF] Morita | Characterization of impact damage resistance of CF/PEEK and CF/toughened epoxy laminates under low and high velocity impact tests[END_REF]. Furthermore, the growth of interfacial cracks will be driven by the different crack tip loading conditions, where the Mode I and Mode II delamination patterns are often considered to be the most important failure modes [START_REF] Boon | A review of methods for improving interlaminar interfaces and fracture toughness of laminated composites[END_REF]. Some studies have focus on the interfacial fracture toughness of the metal-composite adhesive joints [START_REF] Cortés | The fracture properties of a fibre-metal laminate based on magnesium alloy[END_REF][START_REF] Reyes | Mechanical behavior of lightweight thermoplastic fiber-metal laminates[END_REF][START_REF] Abdullah | Interfacial fracture of the fibre-metal laminates based on fibre reinforced thermoplastics[END_REF]. Cortés and Cantwell [START_REF] Cortés | The fracture properties of a fibre-metal laminate based on magnesium alloy[END_REF] indicated that the Mg/CFRP sample offered a significantly higher fracture energy than the Mg/GFPP system, and the fracture energy for Mg/CFRP sample was slightly lower than that of plain CFRP. Reyes and Kang [START_REF] Reyes | Mechanical behavior of lightweight thermoplastic fiber-metal laminates[END_REF] mentioned that the critical strain energy release rate for metal-composite joint consisting of glass fiber reinforced polypropylene and aluminium significantly exceeded that of pure composite. Some studies have been carried out to improve the interlaminar fracture toughness of the metal-composite joints [START_REF] Ning | Mode-II interlaminar fracture toughness of GFRP/Al laminates improved by surface modified VGCF interleaves[END_REF][START_REF] Pan | A new approach to enhancing interlaminar strength and galvanic corrosion resistance of CFRP/Mg laminates[END_REF][START_REF] Arai | Mode I and mode II interlaminar fracture toughness of CFRP laminates toughened by carbon nanofiber interlayer[END_REF]. However, most of researches focus on aluminum-CFRP or magnesium-CFRP hybrid composites. The study on interfacial fracture toughness of steel-CFRP hybrid composites, especially detailed insight into the effect of surface treatment and adhesive on both Mode I and Mode II interfacial fracture toughness, has rarely been reported and is still required for further investigation.

In this study, the Mode I and Mode II interfacial fracture toughness of steel-CFRP hybrid composites prepared by different manufacturing process were investigated, which included the mechanical treatment applied to the steel surface and introduction of adhesive on the interface of the hybrid composites. Moreover, finite element analyses (FEA) were conducted to simulate fracture processes of steel-CFRP hybrid composites under Mode-I and Mode-II loading conditions and the interfacial tensile strength is successfully predicted which is difficult to obtained experimentally.

Material and methods

Materials

The steel substrates used in this study were hot stamping boron steel B1500HS from Baosteel Company in China, which was industrially processed to cold rolled steel plates with a thickness of 1.5 mm. A fastcuring prepreg (Y01-1 from Shenzhen No. 1 Company) with 200 g/m 2 in a twill fabric form was used to fabricate composite reinforcement. The matrix resin consisted of modified epoxy resin and modified imidazole microcapsules as curing agent. The epoxy number and average molecular weight of the matrix epoxy resin were about 0.5 eq/100g and 400 g/mol, respectively. The modified imidazole with 20% modifier was sealed in microcapsules. The epoxy adhesive used in this paper was SW-6 from Shanghai Huayi, a two-component adhesive that contained bisphenol A epoxy resin and modified phenolic amine as curing agent. The epoxy number and average molecular weight of the bisphenol A epoxy resin were about 0.4 eq/100g and 400 g/mol, respectively. The amine number of the curing agent was about 450 mgKOH/g. These two components were mixed accurately according to the weight ratio of 100 (bisphenol A epoxy resin) and 40 (curing agent) under the room temperature. The gelation time was about 2 h at 25 • C. The general physicochemical characteristics of the matrix resin and epoxy adhesive are provided by the manufacturer as shown in Table 1.

The mechanical properties of steel and CFRP were experimentally tested in our previous study [START_REF] Yao | Manufacturing and mechanical properties of steel-CFRP hybrid composites[END_REF], as shown in Table 2.

Preparation of hybrid composites

The fabricated steel-CFRP hybrid composites consisted of a 1.50 mm thick steel layer and a 1.50 mm thick CFRP layer. As shown in Table 3, four types of steel-CFRP hybrid composites were prepared with different manufacturing processes.

Surface preparation of steel substrate

Abrasion or grit blasting has been commonly used to prepare the steel substrate to have rough, fresh and more reactive contact surface areas between the adhesive and substrate. Compared with abrasion, grit blasting is more effective to achieve a coarse and high-energy surface. These two mechanical treatment techniques were employed to prepare the surface of the steel substrate, abrasion (AB) and grit blasting (GB). Mechanical abrasion was carried out on the steel surface with an air sander, and the used sandpaper grit was 80. During grit blasting, 60 mesh white alundum (Al 2 O 3 ) as blasting media was ejected onto the steel surface with a suction-fed, dry grit blasting machine. The pressure was 0.4 MPa with an approach angle of approximately 45 • and a standard distance of 40 mm. After mechanical treatment, the steel substrates were degreased with acetone.

Forming process of hybrid composites

A co-curing compression molding process was used to prepare the hybrid composites. During the curing process, the CFRP prepregs with or without the adhesive were cured and bonded to the steel substrate in one-step. Two types of samples were prepared for comparison. One is the samples (HY-AB and HY-GB) without adhesive between steel and CFRP. For the other one, an epoxy adhesive SW-6 was introduced to the steel/ CFRP interface (HY-ABAD and HY-GBAD). The mixed liquid adhesive was evenly applied to the surface of the steel substrate by a spray gun. The spread quantity was about 60 g/m 2 which was controlled by the spray cycles with constant spread quantity per second. To provide a starter pre-crack notch, Teflon film with 0.5 mm thickness was inserted into steel/CFRP interface at one edge of the hybrid composites. Then six layers of CFRP prepregs with consistent 0 • /90 • in fibers direction were placed on a steel substrate in a mold as shown in Fig. 1. The panels with a size of 290 mm × 190 mm were cured at 150 • C for 5 min in a hot-press machine. The heating rate was set to 5 • C/min and the pressure was kept at 2.5 MPa during the curing process. At last, the hybrid composites were removed from the mold after cooling to below 30 • C with a water cooling system. Finally, four types of steel-CFRP hybrid composites were prepared. All of the test specimens were cut from the hybrid composites prepared above with a CNC Water Jet Cutting Machine possessing a machining accuracy of 0.1 mm. All specimens were stored at the standard laboratory atmosphere of 23 • C and 50-60% relative humidity for 24 h at least before testing, and all tests were conducted at the standard laboratory atmosphere. The average value for at least five specimens was taken as the test result for each case.

Double cantilever beam (DCB) test

According to ASTM D5528, a standard double cantilever beam (DCB) test was employed to evaluate Mode I interfacial fracture toughness. The DCB specimens were 135 mm in length, 25 mm in width with initial crack length of 50 mm. A pair of piano hinge tabs was bonded to the end of the specimens to introduce the load to the specimen during testing. To permit visual crack-tip location, the two sides of the specimen were painted white. The DCB tests were performed in an Instron 5985 test machine under a cross-head speed of 2 mm/min. Load versus displacement of each composite specimen was recorded to measure the interfacial fracture toughness G IC . Location of the crack tip was tracked down regular intervals (every 5 mm) and recorded along with the applied loading and opening displacement at each measured crack extension.

According to the modified beam theory (MBT) introduced in ASTM D5528, the Mode I interfacial fracture toughness was determined as follows:

G IC = 3Pδ 2b(a + Δ) ( 1
)
where P is applied load, δ is load point displacement, b is specimen width, a is delamination crack length, and Δ is the corrective factor for crack length which is determined experimentally by generating a least squares plot of the cube root of compliance, C 1 3 , as a function of delamination crack length (a). The compliance, C, is the ratio of the load point displacement to the applied load, δ/P.

End notched flexure (ENF) test

Mode II interfacial fracture toughness was characterized using an end notched flexure (ENF) test according to the ASTM D7905. The ENF specimens with length of 140 mm, width of 25 mm and initial crack length of 30 mm were cut from the fabricated hybrid composite plates. The ENF test was performed by a universal material testing machine (Instron 5985) under the cross-head speed of 2 mm/min at room temperature. Values of load and displacement were measured to calculate the Mode II interfacial fracture toughness G IIC .

According to the compliance calibration method (CCM) introduced in ASTM D7905, the following expression was used to calculate G IIC :

G IIC = 3mP 2 Max a 2 0 2B (2)
where P Max is maximum load from the fracture test, a 0 is delamination crack length used in the fracture test (30 mm), B is specimen width, and m is the slope obtained from the regression analysis of the compliance, C, versus crack length cubed (a 3 ) of the form: C = A + ma 3 .

Material characteristics

Finally, the detached surface morphology of the specimens after DCB and ENF tests were examined using a laser confocal scanning microscope (VK-X200K).

Finite element model

To simulate crack initiation and propagation processes during DCB and ENF tests, two-dimensional finite element analysis (FEA) models of hybrid composites were built with commercial software ABAQUS, as shown in Fig. 2. The models are made up with three main components: Steel substrate (CPE4I: 4-node bilinear plane strain elements), CFRP laminate (CPE4I), and Steel/CFRP interface (COH2D4: 4-node twodimensional cohesive element.). The CFRP laminate was modeled as orthotropic material by effective properties of engineering constants. The detailed material parameters of steel substrate and CFRP laminates in simulations are shown in Table 2.

Cohesive elements are useful in modeling adhesives and bonded interfaces. A typical traction-separation response was adopted to define the progressive damage and failure in cohesive layers in ABAQUS/ Explicit. The elastic behavior of the cohesive elements before damage initiation can be described as following [START_REF]Abaqus/CAE user's manual DS SIMULIA 2016 providence Rhode Island[END_REF]:

t i = K i δ i , i = n, s, t (3)
where t, and δ represent the nominal traction stress vector and the corresponding separation, respectively, and n, s, and t represent the normal direction and two shear directions. The maximum nominal stress criterion was adopted to define the damage initiation which is shown below [START_REF]Abaqus/CAE user's manual DS SIMULIA 2016 providence Rhode Island[END_REF]:

max { t n t 0 n , t s t 0 s , t t t 0 t } = 1 (4)
where t 0 i (i = n, s, t) are the corresponding maximum traction in each direction. The delamination propagation was described by linear damage evolution based on fracture energy. For separate mode, tractionseparation law mainly depends on the three parameters, i.e., initial stiffness (penalty stiffness, K), maximum traction (interfacial strength, N) and steady-state fracture energy (critical fracture energies, G).

Results and discussion

Mode I interfacial fracture toughness (DCB test)

Typical load-displacement curves of the hybrid composite specimens during DCB tests are shown in Fig. 3a. For each specimen, the load presents linear increase until a critical load is achieved, then the load gradually deceases with the delamination crack propagation. For the specimen with adhesive (HY-ABAD, HY-GBAD), the load fluctuates greatly during decreasing. The calculated delamination resistance curves (R-curves) are shown in Fig. 3b. The calculated Mode-I fracture toughness (G IC ) is basically stable with delamination crack growth. The average fracture toughness of the four specimens is calculated and shown in Fig. 3c, and the standard deviations are indicated by the error bar. With grit blasting treatment on the steel surface, a great increase of G IC for the HY-GB is observed compared with the HY-AB. For the specimen with adhesive, the HY-GBAD also shows higher G IC than the HY-ABAD. This indicates that grit blasting treatment can enhance the interfacial fracture toughness of steel/CFRP composites effectively. With the introduction of adhesive into the steel/CFRP interface, the G IC of the HY-ABAD gains highly improvement compared with that of HY-AB. Moreover, a further increase in the HY-GBAD demonstrates that combine grit blasting treatment on the steel surface and introduction of adhesive into the steel/CFRP interface, a superimposed improvement effect is produced.

Mode II interfacial fracture toughness (ENF test)

Fig. 4a shows typical load-displacement curves of ENF tests for steel-CFRP hybrid composites. For each specimen, the load initially presents linear increase, then shows a nonlinear increase after the yield point of the steel is reached. While the load reaches the peak value, drop occurs due to crack propagating at the critical load. The average Mode-II fracture toughness G IIC for four specimens are calculated and shown in Fig. 4b. It shows similar tendency with the G IC . Grit blasting treatment and introduction of adhesive can enhance the Mode II interfacial fracture toughness of steel/CFRP composites effectively, since the G IIC for the HY-GB and the HY-ABAD are obviously higher than that of the HY-AB. The HY-GBAD which combines grit blasting treatment and introduction of adhesive possesses the highest G IIC , which is much higher than those of HY-GB and HY-ABAD specimen. Thus, combine grit blasting treatment on the steel surface and introduction of adhesive into the steel/CFRP interface, a same superimposed improvement effect is produced on the Mode II interfacial fracture toughness of steel/CFRP composites.

Numerical simulation with the finite element method

As above, under Mode-I loading condition, the fracture toughness G IC of steel/CFRP hybrid composites were obtained experimentally. The interfacial tensile strength N I , which relates to the critical load for fracture, should be tested independently, such as a butt-joint test under tensile load. However, the butt-joint test for hybrid composites is difficult to carry out. Here, by matching the numerical load-displacement curves of DCB tests to the typical experimental ones, the interfacial tensile strengths N I of four steel/CFRP hybrid composites were determined, similar to Li et al. [START_REF] Li | Improvement of interlaminar mechanical properties of CFRP laminates using VGCF[END_REF]. Material properties of cohesive element are listed in Table 4. Note that G IC is the experimental value obtained in Section 3.1, E is the elastic modulus of the adhesive, and the initial stiffness (K) of cohesive elements before crack propagation is automatically calculated by dividing the true thickness of the cohesive elements (0.01 mm). As illustrated in Fig. 5, comparing the four specimens with different G IC , we can find that the maximum predicted loads show the same order of difference. The three numerical curves shown in each subplot are calculated by considering the predicated interfacial tensile strengths N I and 20% higher and lower from that level (N I , 1.2N I , 0.8N I ). With a change of 20% in N I , the maximum predicted load shows small difference, the same with the slope of the curve near maximum load. This indicates that the crack propagation is mainly driven by energy [START_REF] Anyfantis | On the failure analysis of bondlines: stress or energy based fracture criteria?[END_REF]. However, the interfacial tensile strengths N I may be roughly deduced considering both the maximum load and the slope of the curve. As shown in Table 4, compare with 3 MPa for HY-AB, the N I of HY-GB and HY-ABAD are enhanced to 8 MPa and 12 MPa, respectively. With grit blasting treatment and introduction of adhesive, the HY-GBAD shows lager strength with a value of 20 MPa. Therefore, both For Mode-II loading condition, the interfacial shear strengths obtained from single-lap shear tests in our previous study [START_REF] Yao | Manufacturing and mechanical properties of steel-CFRP hybrid composites[END_REF] were used as N II , as list in Table 5. G is the shear modulus of the adhesive. The comparison between numerical results of ENF tests and experimental ones (closest to the mean) is shown in Fig. 6. Since the experimental values of the interfacial shear strength N II and G IIC were directly used in the simulations, the good consistency between experimental and numerical results confirms the appropriateness of the ENF model.

Conclusively, the above results verify the appropriateness of the finite element models, and the appropriate values of each FEM parameters of the hybrid composites are clarified, which are helpful for the prediction of mechanical properties of complex structure. Moreover, both the numerical simulations and the experiments verify the improvement effect of grit blasting treatment and introduction of adhesive on the Mode-I and Mode-II interfacial mechanical properties of steel-CFRP hybrid composites.

Failure mode analysis

Fig. 7 presents the digital photographs of the detached surfaces after DCB tests. As shown in Fig. 7a, the failure of HY-AB mainly occurs at the steel/CFRP interface. The failure mode of the specimen is mainly adhesive failure between steel and epoxy. With grit blasting treatment, the HY-GB exhibits a mixed fracture mode, including fiber/epoxy interfacial failure and adhesive failure, since the CFRP can be observed on the detached steel surface with some matrix resin at where the fibers interweave, as shown in Fig. 7c. For the specimen with adhesive (HY-ABAD), the presence of adhesive cohesive failure can be confirmed with large area of residual adhesive on the detached steel surface, as shown in Fig. 7b. The specimen exhibits a mixed fracture mode, including adhesive cohesive failure and adhesive failure between adhesive and steel, accompanied with a degree of fiber/epoxy interfacial failure. With grit blasting treatment and introduction of adhesive, fiber/epoxy interfacial failure becomes the main failure mode of HY-GBAD. As shown in Fig. 7d, there is no steel surface exposed with CFRP and adhesive covered on it.

The failure modes of hybrid composites during ENF tests are much different with that during DCB tests. As shown in Fig. 8, all the four specimens mainly exhibit adhesive failure between adhesive and steel. Epoxy resin is observed on the grit blasted steel surface, indicating epoxy and adhesive cohesive failure occurs for HY-GB and HY-GBAD. The fiber/epoxy interfacial failure mode is rarely seen in Mode II loading conditions.

As we discussed in our previous study [START_REF] Yao | Manufacturing and mechanical properties of steel-CFRP hybrid composites[END_REF], grit blasting with the grit of 60 mesh would produce much rougher steel surfaces when compared with abrasion with sandpaper of 80 mesh. The surface roughness of the steel surface after abrasion and grit blasting are 1.6 mm and 10.0 mm, respectively. The thickness of the interphase region between steel and CFRP laminate (adhesive layer) was about 10 μm [START_REF] Yao | Manufacturing and mechanical properties of steel-CFRP hybrid composites[END_REF]. Rather than the thickness increase of the steel/CFRP interphase region and tougher property of the introduced adhesive, the steel/CFRP interfacial properties should mainly depend on the interface bonding of steel and adhesive. The good wetting of the adhesive on the steel substrate provides close contact between them and forms a good bond. The rougher surface generates by grit blasting significantly increase the interface area between steel and adhesive, which should contribute to the improvement of Mode I fracture toughness. The ravine features of steel surface can interlock with the epoxy, which may cause the cohesive failure of epoxy under shear loading condition and enhances the Mode II fracture toughness through mechanical interlock.

In this work, the values of Mode II fracture toughness of hybrid composites are similar with that of Mode I fracture toughness. However, the failure modes of hybrid composites under Mode I and Mode II loading conditions show a big difference, which indicates that the failure modes in the hybrid composites are governed by different crack-tip loading conditions. With grit blasting treatment and introduction of adhesive, the main failure mode of HY-GBAD under Mode I loading condition is fiber/epoxy interfacial failure. This indicates that the Mode I interfacial fracture toughness of the steel/CFRP interface exceeds that of the plain CFRP. However, under Mode II loading condition, the HY-GBAD mainly exhibit adhesive failure between adhesive and steel, similar with other three specimens. There are still large gaps for the Mode II interfacial fracture toughness of the steel/CFRP interface compared with that of the plain CFRP. Some studies reported that for fiber reinforced polymers, there was a degree of difference between the G IC and G IIC , and the G IIC was about twice higher than G IC [START_REF] Cortés | The fracture properties of a fibre-metal laminate based on magnesium alloy[END_REF][START_REF] Arai | Mode I and mode II interlaminar fracture toughness of CFRP laminates toughened by carbon nanofiber interlayer[END_REF][START_REF] Reyes | The mechanical properties of fibre-metal laminates based on glass fibre reinforced polypropylene[END_REF]. This may attribute to the bridging fibers connecting two opening surfaces for fiber reinforced polymers. Therefore, although the G IC of the hybrid composites is higher than that of CFRP, the G IIC of the hybrid composites is lower than that of CFRP, which results in that the steel/CFRP adhesive failure is the main failure mode of the hybrid composites under Mode II loading condition. Moreover, due to the structural asymmetry and material heterogeneity of hybrid composites, the interfacial fracture toughness from the DCB tests is in Mode I/II mixing at the crack tip [START_REF] Hua | Interlaminar fracture toughness of GLARE laminates based on asymmetric double cantilever beam (ADCB)[END_REF][START_REF] Zakaria | Effects of aluminum surface treatments on the interfacial fracture toughness of carbon-fiber aluminum laminates[END_REF][START_REF] Sundararaman | An unsymmetric double cantilever beam test for interfacial fracture toughness determination[END_REF]. The energy release rates in the initial crack lengths are mainly contributed by Mode I. With increasing the crack length, higher title angle of the specimen causes more contribution of Mode II on the fracture and increases the overall fracture toughness [START_REF] Zakaria | Effects of aluminum surface treatments on the interfacial fracture toughness of carbon-fiber aluminum laminates[END_REF]. Since the G IC has a great influence on the fitting of the load-displacement curves, the deduced interfacial tensile strength N I may deviate from the real value. To eliminate the effect of shear loading at the crack front and assuring pure mode-I loading, the flexural rigidity of the CFRP should be equal to the flexural rigidity of the steel [START_REF] Wang | A review of experimental and theoretical fracture characterization of bi-material bonded joints[END_REF].

Conclusions

The paper studied the interfacial fracture toughness of co-cured steel-CFRP hybrid composites with the different manufacturing process and surface treatment. The results show that the Mode I and Mode II interfacial fracture toughness of the hybrid composites can be improved by using a grit blasting surface treatment on steel and introducing an adhesive layer at the steel/CFRP interface. The improved hybrid composites mainly show fiber/epoxy interfacial failure of CFRP under Mode I loading conditions, while for Mode II loading, it mostly exhibits adhesive failure of steel/CFRP interface. This may attribute to the higher G IC and lower G IIC of the hybrid composites, compared with the plain CFRP. Furthermore, the steel/CFRP interfacial tensile strength is predicted by finite element analysis, the good consistency between experimental and numerical results confirms the appropriateness of finite element models, and the appropriate values of each FEM parameters of the hybrid composites are clarified.

Fig. 1 .

 1 Fig. 1. Schematic of stacking arrangement of steel-CFRP composites in a mold.

Fig. 2 .

 2 Fig. 2. FEA model of steel/CFRP hybrid composites for (a) DCB and (b) ENF tests.

Fig. 3 .

 3 Fig. 3. (a) Typical load-displacement curve of DCB tests, (b) Calculated delamination resistance curves (R-curves) and (c) Mode-I interfacial fracture toughness (G IC ) for steel-CFRP hybrid composites with different manufacturing processes.

Fig. 4 .

 4 Fig. 4. (a) Typical load-displacement curve of ENF tests and (b) Calculated Mode-II fracture toughness (G IIC ) for steel-CFRP hybrid composites with different manufacturing process.

Fig. 5 .

 5 Fig. 5. Numerical and experimental load-displacement curves of DCB tests for steel-CFRP hybrid composites with different manufacturing processes.

Fig. 6 .

 6 Fig. 6. Numerical and experimental load-displacement curves of ENF tests for steel-CFRP hybrid composites with different manufacturing process.

Fig. 7 .

 7 Fig. 7. Digital photographs of the detached surfaces after DCB tests for the steel-CFRP hybrid composites with different manufacturing process: (a) HY-AB, (b) HY-ABAD, (c) HY-GB, (d) HY-GBAD.

Fig. 8 .

 8 Fig. 8. Digital photographs of the detached surfaces after ENF tests for the steel-CFRP hybrid composites with different manufacturing process: (a) HY-AB, (b) HY-ABAD, (c) HY-GB, (d) HY-GBAD.

Table 1

 1 General characteristics of the matrix resin and epoxy adhesive.

		Type	Tensile strength (25 • C) [MPa]	Tensile modulus (25 • C) [MPa]	Elongation (25 • C) [%]	Poisson's coefficient
	Matrix resin Epoxy adhesive	Film Liquid	91 63	3500 1907	5.2 7.5	0.34 0.34

Table 2

 2 Material properties of steel and CFRP. E 22 = 48, E 33 = 3.5 G 12 = 3.66, G 13 = G 23 = 4.59 Poisson's ratio ν = 0.3 ν 12 = 0.37, ν 13 = ν 23 = 0.28

	Properties	Steel	CFRP
	Modulus (GPa)	E = 235	E 11 =

Table 3

 3 Sequence of manufacturing process for hybrid composites.

	Code	Surface preparation	Forming process
	HY-AB HY-ABAD HY-GB HY-GBAD	Abrasion (#80 grit) Abrasion (#80 grit) Grit blasting (#60 grit) Grit blasting (#60 grit)	Co-cured without adhesive Co-cured with adhesive Co-cured without adhesive Co-cured with adhesive

Table 4

 4 Material properties of cohesive element in DCB model.

	Specimen	E [MPa]	G IC [kJ/m 2 ]	N I [MPa]
	HY-AB HY-ABAD HY-GB HY-GBAD	3500 1907 3500 1907	0.100 0.825 0.835 1.161	3 8 12 20

Table 5

 5 Material properties of cohesive element in ENF model.

	Specimen	G [MPa]	G IIC [kJ/m 2 ]	N II [MPa]
	HY-AB HY-ABAD HY-GB HY-GBAD	1306 711 1306 711	0.096 1.01 0.843 1.733	17.56 22.01 18.68 23.69

Acknowledgements

The authors gratefully acknowledge the financial support from the National Nature Sciences Foundation of China (U1809218, U1864211), Natural Science Foundation of Zhejiang Province (LQ20E030004), Ningbo Key Projects of Science and Technology Innovation 2025 Plan (2019B10118, 2020Z030, 2020Z116), Instrument Development Project of CAS (YJKYYQ20200029), and CAS President's International Fellowship Initiative (2020DE0011).


Bakhyt T Matkarimov
email: bmatkarimov@nu.edu.kz

Murat K Saparbaev
email: murat.saparbaev@gustaveroussy.fr

Chargaff's second parity rule lies at the origin of additive genetic interactions in quantitative traits to make natural selection possible

ABSTRACT

A major challenge of modern biology is to understand how random genetic variation causes phenotypic variation in quantitative traits. Chargaff's Second Parity Rule (CSPR) is a statistical property of cellular genomes defined as near exact equalities G≈C and A≈T within single DNA strand. Analysis of mutation spectra inferred from single nucleotide polymorphisms (SNPs) observed in human and mice populations reveals near exact equalities of the reverse complementary mutations, indicating that genetic variations obey CSPR. Furthermore, the nucleotide compositions of coding sequences are statistically interwoven via CSPR since pyrimidine bias at 3 rd codon position compensates purine bias at 1 st and 2 nd positions. Due to this inter-dependence both synonymous and non-synonymous mutations are subject to natural selection. Based on the Fisher's infinitesimal model, we propose that accumulation of a sufficient number of the reverse complementary mutations results in a continuous trait variation due to small additive effects of statistically inter-linked genetic alterations. Therefore, additive genetic interactions can be inferred as the statistical entanglement of nucleotide compositions of different genetic loci. CSPR challenges the neutral theory of molecular evolution, because all random mutations participate in variation of the traits. The sequence of a gene is interwoven with many other loci in a genome, each of which making infinitely small contribution to a trait variation in contactless manner.

Intra-strand DNA symmetry and genetic variations in cellular organisms. Chargaff's Second Parity Rule (CSPR), also referred to as intra-strand DNA symmetry, was incidentally determined by measuring nucleotide composition of single-stranded genomic DNAs [START_REF] Rudner | Separation of microbial deoxyribonucleic acids into complementary strands[END_REF] and later rediscovered and confirmed by others [START_REF] Prabhu | Symmetry observations in long nucleotide sequences[END_REF][START_REF] Forsdyke | Relative roles of primary sequence and (G + C)% in determining the hierarchy of frequencies of complementary trinucleotide pairs in DNAs of different species[END_REF][START_REF] Sueoka | Intrastrand parity rules of DNA base composition and usage biases of synonymous codons[END_REF][START_REF] Lobry | Properties of a general model of DNA evolution under no-strand-bias conditions[END_REF][START_REF] Qi | Compositional symmetries in complete genomes[END_REF][START_REF] Baisnée | Why are complementary DNA strands symmetric?[END_REF] . Noteworthy, intra-strand DNA symmetry can be extended to any pair of reverse complementary short oligonucleotides for a sufficiently long DNA sequence (>100 kb), it means the near exact equalities of reverse complementary pairs of monoand oligo-nucleotides in one strand of cellular DNA [START_REF] Prabhu | Symmetry observations in long nucleotide sequences[END_REF][START_REF] Fickett | Base compositional structure of genomes[END_REF][START_REF] Albrecht-Buehler | Asymptotically increasing compliance of genomes with Chargaff's second parity rules through inversions and inverted transpositions[END_REF] . CSPR holds for genomes of cellular organisms (including eukaryotic, bacterial and archaeal chromosomes) and double-stranded DNA viruses, whereas, organellar genomes (mitochondria and plastids) smaller than ~20-30 kbp, singlestranded viral DNA genomes and any type of RNA genome do not obey CSPR [START_REF] Mitchell | A test of Chargaff's second rule[END_REF] . Intriguingly, intra-strand symmetry is maintained in genomic DNA of all cellular organisms despite billions years of divergent evolution of the species from one common ancestor. When regarding at the scale of a chromosome or entire genome, cellular organisms obey CSPR with small deviations less than 1% from mononucleotide parity rule [START_REF] Mitchell | A test of Chargaff's second rule[END_REF] . It remains unknown: what are the mechanisms that force cellular genomes to adapt CSPR and why do genomes of certain non-cellular organisms have broken intra-strand DNA symmetry. Previously it has been proposed that the skewed nucleotide composition of vertebrate mitochondrial genomes is due to asymmetric DNA strand inheritance [START_REF] Matkarimov | DNA Repair and Mutagenesis in Vertebrate Mitochondria: Evidence for Asymmetric DNA Strand Inheritance[END_REF] .

Here, we sought to identify a biological role of intra-strand DNA symmetry in cellular organisms for this we explore a link between CSPR and continuous variation observed in complex traits of higher eukaryotes. Quantitative or complex traits in plants and animals are characterized by measurable and continuous phenotypic variation within a population due to underlying genetic variability and environmental influence. Complex traits such as human height do not follow simple Mendelian inheritance patterns, since their phenotypic variations cannot be explained by the segregation of a single genetic factor. In 1919 Ronald Fisher showed that the variation in continuous traits could be explained by Mendelian inheritance laws if an infinite number of genetic loci control the trait and that each locus contributes additively to the trait [START_REF] Fisher | The Correlation between Relatives on the Supposition of Mendelian Inheritance[END_REF] . In his work, Fisher introduced the infinitesimal model, also known as polygenic model, based on the idea that variation in a quantitative trait is influenced by an infinitely large number of genes, each of which with an infinitely small (infinitesimal) effect on the phenotype. Importantly, quantitative traits are far more common as compared to simple Mendelian traits; moreover, even the monogenic traits can exhibit phenotypic heterogeneity in organisms with the same genotype because of complex genetic interactions such as incomplete penetrance, variable expressivity and pleiotropy. Since, quantitative traits can be extremely polygenic, they are examined using statistical techniques such as genome-wide association studies (GWAS) rather than classical molecular biology methods [START_REF] Anthony | Introduction to genetic analysis[END_REF] .

Infinitesimal model provides a basis for statistical modelling and selection theory, which are used in plant and animal breeding programs with great success. Despite this, the model remains a theoretical abstraction, since in reality all living organisms have finite size genomes each containing quite limited number of genes. Prior to the high-throughput DNA sequencing era, it was impossible to verify the infinitesimal model for a quantitative trait in a genetically heterogeneous population. Variants discovered through GWAS accounted for only a small percentage of predicted heritability and this led to the missing heritability problem [START_REF] Manolio | Finding the missing heritability of complex diseases[END_REF] . With advances in DNA genotyping technologies the problem was in part resolved by demonstrating that most of the heritability could be accounted for by common single-nucleotide polymorphisms (SNPs) missed by GWAS, because their effect sizes fell below genome-wide statistically significant thresholds [START_REF] Claussnitzer | A brief history of human disease genetics[END_REF][START_REF] Purcell | Common polygenic variation contributes to risk of schizophrenia and bipolar disorder[END_REF][START_REF] Yang | Common SNPs explain a large proportion of the heritability for human height[END_REF] . Furthermore, it has been demonstrated that for most of complex traits the majority of the SNPs-associated heritability is spread uniformly all over the genome [START_REF] Loh | Contrasting genetic architectures of schizophrenia and other complex diseases using fast variance-components analysis[END_REF][START_REF] Shi | Contrasting the Genetic Architecture of 30 Complex Traits from Summary Association Data[END_REF][START_REF] O'connor | Extreme Polygenicity of Complex Traits Is Explained by Negative Selection[END_REF] .

Recent studies estimated that more than 10,000 or even more than 100,000 causal variants contribute to a quantitative trait in general, which is in remarkable consistency with infinitesimal model [START_REF] O'connor | Extreme Polygenicity of Complex Traits Is Explained by Negative Selection[END_REF][START_REF] Zhang | Estimation of complex effect-size distributions using summary-level statistics from genome-wide association studies across 32 complex traits[END_REF] .

While many common variants involved in quantitative traits have been identified, the molecular mechanisms through which they contribute to phenotypic variation remain poorly understood [START_REF] Albert | The role of regulatory variation in complex traits and disease[END_REF] . For example, the infinitesimal model states that each genetic locus contributes additively to the trait, while it remains unclear how genetic interactions can act in additive manner and what are the molecular mechanisms of this complementarity. Recently, Boyle and colleagues when analyzing GWAS datasets of the complex traits including human height, schizophrenia, rheumatoid arthritis and Crohn's disease found that the SNPs associated with traits were evenly distributed all over the genome, rather than clustered within the trait-specific genes [START_REF] Boyle | An Expanded View of Complex Traits: From Polygenic to Omnigenic[END_REF] . To explain these observations, they proposed the concepts of "core" genes (limited number of disease-specific genes with established biological roles) and "omnigenic" model in which majority of active genes, rather than a specific set of core genes, can affect every complex traits. For example, the total set of expressed genes in an affected tissue or a cell in general outnumber "core" genes by 100:1 or more, consequently the sum of small effects across peripheral genes can far exceed the genetic contribution of variants of core genes because of the highly interconnected cell regulatory networks [START_REF] Boyle | An Expanded View of Complex Traits: From Polygenic to Omnigenic[END_REF][START_REF] Sinnott-Armstrong | GWAS of three molecular traits highlights core genes and pathways alongside a highly polygenic background[END_REF] . In support to this, GWAS signals for quantitative traits and disease-associated SNPs are enriched in predicted gene regulatory elements and in open chromatin that is active in cell types relevant to disease, respectively [START_REF] Farh | Genetic and epigenetic fine mapping of causal autoimmune disease variants[END_REF][START_REF] Finucane | Partitioning heritability by functional annotation using genomewide association summary statistics[END_REF][START_REF] Wood | Defining the role of common variation in the genomic and biological architecture of adult human height[END_REF][START_REF] Kundaje | Integrative analysis of 111 reference human epigenomes[END_REF] .

High polygenic character of quantitative traits and common diseases might also be explained by negative selection towards large effect mutations in the critical genes and loci that play biologically important functions. In support to this idea, statistical modelling of 33 complex traits reveals that heritability is spread more evenly among common variants than among SNPs in functionally important regions due to selective constraints [START_REF] O'connor | Extreme Polygenicity of Complex Traits Is Explained by Negative Selection[END_REF] . While researchers were able to identify many genetic factors, such as constitutively active peripheral genes, involved in complex traits, determining the specific molecular mechanisms, through which they contribute to phenotypic variation, remains a major challenge [START_REF] Sinnott-Armstrong | GWAS of three molecular traits highlights core genes and pathways alongside a highly polygenic background[END_REF] .

Results and Discussion

Continuous variation in complex traits enables natural selection. Quantitative traits exhibit continuous normal distribution of phenotypic variations because of their polygenic, multi-locus nature. According to the infinitesimal model proposed by Fisher, continuous phenotypic variation of a quantitative trait, influenced by infinitely large number of genetic loci in a population, have normal distribution (Fig. 1a) [START_REF] Nelson | A century after Fisher: time for a new paradigm in quantitative genetics[END_REF] . Here we suggest that this bell-shaped distribution also reflects differential selective pressure or fitness of each individual in the population. For example, when considering human height, individuals with heights close to average experience less selective pressure and present in higher numbers in the population since they are better adapted, whereas shortest and tallest individuals endure high selective pressure and thus present in much lower numbers. Similar in multicellular organism, population of somatic cells with mean variations of a trait undergo less selective pressure and present in higher numbers in a given tissue or organ since they are better fit to an internal body environment, whereas cells with extreme phenotypes undergo high selective pressure and present in much lower frequencies because they are less fit. However, as shown in Fig. 1b, variations in the external environment and organism homeostasis can change the direction of selection pressure and as a consequence organisms and somatic cells, respectively, exhibiting positive (+) trait variation would have advantage and this would shift common variants or allele frequency distribution to the right of the abscissa axis. Cell as a part of highly complex multicellular organism is experiencing a constant selective pressure, because of the internal environment within a body that impose severe restraints on the somatic cells. Multiple factors, among many other such as limited availability of oxygen and nutrients, immune system surveillance, cell-cell contacts, cell-signaling factors that tightly control growth and proliferation of individual cells within a body, all together create selective pressure on a population of somatic cells in order to integrate every part and function of the body as a coherent whole. In fact, cell differentiation, in which changes in the gene transcription programs occurs through epigenetic mechanisms without alterations in the primary DNA sequence, can be regarded as a way for a somatic cell to adapt to continuously changing inner body environment of the complex multicellular organisms during their development and aging. On the other side, random somatic mutations that inevitably accumulate after each cell division, are subject to the strong negative selection pressure that eliminates cells with deleterious and cancer prone mutations.

Somatic mutagenesis is a random, stochastic process and majority of mutations observed in higher eukaryotes occur in non-coding regions of the genome and thus may evade the strong selective pressure. It is generally agreed that in cancer only a few (driver) mutations strongly influence its progression, with the remaining (passenger) mutations being neutral or slightly deleterious [START_REF] Vogelstein | The Path to Cancer -Three Strikes and You're Out[END_REF] .

Recent Pan-Cancer Analysis of Whole Genomes (PCAWG) variant dataset of the 43,778,859 single nucleotide variants (SNVs) reveal only thousands of mutations that can be identified as drivers, with mean of 5 driver mutations per tumor, whereas, the remaining 99.99% of SNVs have been referred as passenger variants, with no known function and effect on the fitness 31 .

Nevertheless, another analysis of PCAWG variant dataset suggest that combined effect of passenger mutations together with undetected weak drivers, can provide additional 12% effect for identifying cancer phenotypes, beyond driver mutations [START_REF] Kumar | Passenger Mutations in More Than 2,500 Cancer Genomes: Overall Molecular Functional Impact and Consequences[END_REF] .

Despite huge technological progress in DNA sequencing and analysis of big data, yet fundamental gaps remain in our understanding of how normal cells evolve into tumor cells. It is assumed that the absolute majority of somatic and passenger mutations in normal and cancer cells, respectively, are either neutral or slightly deleterious. However, this assumption is challenged by several important observations. The existence of genetic interaction networks between multiple genes on a genome-wide scale that can determine phenotypic variability of the observable traits including the expression of monogenic traits such as rare, highly penetrant, deleterious gene mutations. It has been shown that common genetic variants can critically contribute to incomplete penetrance of severe Mendelian diseases, and that not all individuals with a given genotype (mutant allele) display a corresponding phenotype [START_REF] Chen | Analysis of 589,306 genomes identifies individuals resilient to severe Mendelian childhood diseases[END_REF] . This phenomenon was observed also on other model organisms (reviewed in [START_REF] Goldstein | The complex role of genetic background in shaping the effects of spontaneous and induced mutations[END_REF] ). Numerous studies have convincingly demonstrated that Calorie Restriction (CR), as compared to control Ad Libitum (AL) regimen, dramatically delays spontaneous tumor development in laboratory animals [START_REF] Chen | Effect of Intermittent versus Chronic Calorie Restriction on Tumor Incidence: A Systematic Review and Meta-Analysis of Animal Studies[END_REF] . This dietary regimen changes the body's homeostasis in a way that might reduce the fitness of cancer-prone cells as compared to normal cells and prevents their propagations in normal tissues. Here, we propose that accumulation of sufficient number of random mutations in somatic cells inevitably results in the phenotypic variations owing to small additive genetic interactions. Trait variations in somatic cells such as glucose uptake and proliferation rate are subject to natural selection. Therefore, CR regimen would favor the growth of cells with lower proliferation rate and glucose consumption, whereas, precancerous cells would be under increased negative selection pressure and this in turn would prevent their propagation. Thus, we suggest that common variants and passenger mutations can influence the phenotypic expression of a simple Mendelian trait such as a cancer-driver mutation by uncovering its complex polygenic character. Below, we will analyze the role of intra-strand DNA symmetry in genetic variations and phenotypic diversity.

Intra-strand DNA symmetry and genetic variations in cellular organisms. CSPR can be interpreted as the law of large numbers which states that, as the number of identically distributed, randomly generated variables increases, their sample mean approaches their distribution mean.

Therefore, intra-strand DNA symmetry can be considered as a general statistical property of cellular genomes composed of independently segregating replicons which is maintained over the evolutionary history of life. This means that a sufficiently large number of genetic variations in a population have a statistical tendency in that every single random mutation must be "compensated" by another inverse complementary mutation elsewhere in the genome. This mechanism of compensation manifests itself through the effect of paired reverse-complementary mutations (e. g. C→T pairs with G→A) occurring in one DNA strand that enables to maintain the intra-strand DNA symmetry within entire segregating replicon(s). As shown in Table 1, analysis of the patterns of spontaneous mutations in trinucleotide sequence context inferred from single nucleotide polymorphisms (SNPs) observed in human and mice populations revealed that in single DNA strand of human and mice genomes, the number (V) of X↔Y variations in triplet NXN (referred to here as VNXN↔NYN) is nearly equal to the number of X'↔Y' variations occurring in reverse complementary triplet N'X'N' on the same DNA strand (referred to here as VN'X'N'↔N'Y'N', where, V is for total number of a variation, N is any nucleotide, X and Y are different versions of a nucleotide, N', X' and Y' are corresponding reverse complementary counterparts). For example, the number of SNP AGT↔AAT, referred to here as VAGT↔AAT, observed in human population, is 2.375.298 (100%), which is nearly equal to the number of SNP of its reverse complementary version VACT↔ATT, observed 2.367.625 times (99.7%). Similar statistical parities are observed in mice populations there VAGT↔AAT ≈ VACT↔ATT (384.516 (100%) and 384.248 (99.99%) times, respectively) with less than 1% difference (Table 1 andS1 Furthermore, analysis of small insertion/deletion (indel) polymorphism in human population revealed near exact equalities in the number of indels between 18 out of 32 reverse complementary pair of trinucleotides (Table S2). These trinucleotides have sequence context in which a base corresponding to indel is different from the neighboring base (e.g. VTΔC↔TGC ≈ VGΔA↔GCA). It should be noted that for 14 out of 32 pairs, the correct pairwise alignment was not possible because of sequence context particularity. These trinucleotides have sequence context in which a base corresponding to indel is the same as the neighboring base (e.g. VTΔG↔TTG ≠ VCΔA↔CAA). These observations indicate that during evolution the accumulation of random mutations in a population of cellular organisms, represented here in the form of SNPs and indels, is necessary and sufficient condition to drive and maintain chromosomal DNA in the state of intra-strand symmetry equilibrium. Thus, the stochastic processes such as random mutations can lead genome evolution towards the symmetry of nucleotide compositions and this should be considered as one of the major force of evolution, which provides great selective advantage to cellular organisms.

Statistical inter-dependence of the nucleotide compositions is revealed by a bias at 3 rd position of codon. The analysis of mutation spectra inferred from SNPs suggest that CSPR is an essential property of cellular organisms, which restrains the patterns of random genetic mutations to make them assembled in the pairwise, statistically interwoven manner. Local deviations from intra-strand DNA symmetry in the majority of coding DNA sequences (CDSs), referred as to Szybalski's rule [START_REF] Szybalski | Pyrimidine clusters on the transcribing strand of DNA and their possible role in the initiation of RNA synthesis[END_REF] (when purines exceed pyrimidines), suggest that at the gene level, directional mutation pressures and selective constraints counteracts the homogenizing force of CSPR. To examine this closely, we measured the deviations from intra-strand DNA symmetry of the nucleotide composition at each of three positions of a codon in bacteria and human CDSs and presented data in the graph in which each codon position is plotted according to its purine/pyrimidine (Pur/Pyr or G/C and A/T) ratio. As shown in Fig. 2a, when measuring Pur/Pyr ratio for all three positions of a codon together, the nucleotide compositions of each triplet (denoted as black squares) in bacterial proteomes exhibit relatively weak deviations from CSPR with G/C and A/T ratio nearly equal to one (≈ 1), as compared to that of single codon positions (denoted as colored squares). Remarkably in absolute majority of bacteria, 1 st position of a codon (red square) has highly enriched purine content (Pur/Pyr ratio ≥ 1.5), while positions 2 and 3 (blue and green squares, respectively) contain increased C and T content (G/C and A/T ratio ≤ 1), respectively, indicating that the nucleotide composition at 2 nd and 3 rd positions of a codon compensate very strong bias in the purine content at 1 st position of a codon. As a result, when all three positions of a codon are taken into account, we observe more symmetry in the nucleotide compositions of triplets, which in turn dramatically reduces the deviations of CDSs from CSPR at a gene level in bacteria. In human proteome we can also observe the phenomenon of compensation of purine bias in the nucleotide composition at 1 st position of a codon by that of 2 nd and 3 rd positions, which results in Pur/Pyr ratio ≈ 1 for the nucleotide composition of each triplet (Fig. 2b). Two last codon positions have enriched pyrimidine content (>50%) which is opposite to that of 1 st position of a codon, thus again the Pur/Pyr content at 2 nd and 3 rd positions compensate the strong bias from intra-strand DNA symmetry of the nucleotide composition at 1 st position of a codon in human CDSs. Noteworthy, due to redundancy of the genetic code, mutations at the 3 rd position of a codon often results in a synonymous change of amino acid sequence of the protein, which explains lower evolutionary conservation of this position as compared to first two positions of a codon [START_REF] Kimura | On some principles governing molecular evolution[END_REF] . Mutations at 1 st and 2 nd positions of a codon in majority of cases lead to non-synonymous changes in the protein sequence, which are subject to strong selection pressure, therefore, the first two positions of a codon are much more functionally constrained, as compared to 3 rd one. Nevertheless, our data indicate that variations in the nucleotide composition at 3 rd position of a codon exhibit strong statistical entanglement or inter-dependency with that of 1 st and 2 nd positions, owing to the law of large numbers that maintains CSPR at genome level.

Based on these observations, we suggest that the functional constraints and selective pressure towards first two codon positions, also act on 3 rd codon position, owing to the statistical entanglement of their nucleotide compositions. As a result, an increase in purine content at 3 rd position of codons due to the synonymous transversion mutations would be under strong selection pressure. Indeed, the 3 rd codon position has a much higher transition/transversion mutation bias than the 1 st and 2 nd codon position suggesting that transversions are disproportionally selected against at the last position of a codon [START_REF] Bofkin | Variation in evolutionary processes at different codon positions[END_REF] . Thus, under intra-strand DNA symmetry equilibrium of a gene sequence, both synonymous and non-synonymous mutations in a CDS would be under selective pressure because of their inter-dependence.

Taken together these observations suggest that the synonymous mutations at 3 rd position of a codon, when they accumulate in sufficient number, are neither neutral, nor nearly neutral, but subject to natural selection to the similar extent as the mutations in 1 st and 2 nd codon positions. Hence, by virtue of CSPR, random mutations occurring in cellular genomes at various genetic loci over generations become statistically interwoven and this makes them all subject to natural selection. Moreover, when considering genetic loci such as junk DNA in higher eukaryotes [START_REF] Orgel | Selfish DNA: the ultimate parasite[END_REF] , which are seemingly useless and have no known function, their nucleotide compositions are statistically interwoven with that of functional regulatory and coding DNA sequences, as a consequence, natural selection can act on the genetic variations in both functional and junk DNA. Owing to CSPR, accumulation of mutations in junk DNA will exert increasing selective pressure on the mutation pattern in functional (regulatory and coding) DNA sequences, forcing natural selection act on seemingly functionless DNA regions.

Intra-strand DNA symmetry and the origin of additive genetic interactions. CSPR acts at full strength when a sufficiently big number of random mutations are taken into consideration, for example in a large set, the mutations can be assembled as pairs of reverse complementary matches (see Table 1). Whereas small number of random mutations may not always exhibit the expected intra-strand DNA symmetry, nevertheless with each additional mutation the probability that all observed mutations would be in accordance with CSPR increases. Consequently, when the number of mutations occurring in various genetic loci increases in time dependent manner, they would approach intra-strand DNA symmetry and exhibit strong statistical entanglement. For that reason, increasing number of mutations in non-coding DNA and peripheral gene networks should be compensated in part by the reverse complementary mutations in coding DNA and core gene network, respectively, and vice versa, so that the nucleotide compositions of an entire chromosome comply with CSPR. Furthermore, the homogenizing force of CSPR acts also at local gene level, over the course of multiple generations the nucleotide composition of a CDS should be approaching closely the intra-strand DNA symmetry, despite structural and functional constraints on the evolution of a protein (Fig. 2, black squares).

Above analysis of SNPs and biases at different codon positions further demonstrate that intra-strand DNA symmetry is a general property of cellular organisms that underlies statistical inter-dependence of the nucleotide compositions of different genetic loci and patterns of spontaneous mutations. Hence, we suggest that this statistical entanglement provides a longsearched mechanism of the additive genetic interactions, which constitute the basis of continuous variation in expression of the traits. According to Fischer's infinitesimal model, continuous phenotypic variation in a quantitative trait is achieved through additive genetic interactions of an infinitely large number of genetic loci [START_REF] Fisher | The Correlation between Relatives on the Supposition of Mendelian Inheritance[END_REF] . In general, the effect of a single mutation is infinitesimally small and cannot be measured experimentally. However, the cumulative effect of many such mutations become visible and can be measured, since the mutations become statistically entangled in the course of evolution, and their accumulation give rise to additive genetic effect. Here, we propose that the additive genetic effect is achieved through intra-strand complementarity of the nucleotide compositions of diverse genetic loci participating in a trait variation. The nucleotide compositions of coding DNA sequences and regulatory elements (such as promoters and origins of replication) obey CSPR. In higher eukaryotes these highly conserved DNA sequences are statistically interwoven with other sites in a genome, which in their majority are rather non-coding and non-conserved. Consequently, when a sufficient number of the genetic variations in noncoding DNA sequences are accumulated, this would result in continuous variation of a trait via a modulation of the expression of highly conserved DNA sequences.

Because of CSPR, which can be derived from the law of large numbers, random genetic mutations in a cell or an organism have a statistical tendency to accumulate in pairwise manner in which pair of reverse complementary SNPs present in near equal numbers VX↔Y ≈ VX'↔Y' or VNXN↔NYN ≈ VN'X'N'↔N'Y'N' (e.g. the numbers of G↔A or TGC↔TAC SNPs in a population are near equal to that of C↔T or GCA↔GTA SNPs, respectively (VG↔A ≈ VC↔T or VTGC↔TAC ≈ VGCA↔GTA)). This means that accumulation of C→T mutations would be accompanied with nearly exact number of G→A mutations in the same DNA strand of a chromosome distributed randomly anywhere in a genome. As shown in Fig. 3, we propose that accumulation of the reversecomplementary mutations results in additive genetic effect, which under directional selection would shift the phenotype towards one end of a trait spectrum. The normal distribution of quantitative a trait with mean value of a trait (x̄) at the extremes of X-axis. Note that the distance from core gene is plotted in additional X-axis at the bottom. Accumulation of the statistically inter-dependent mutations in the CSPR-mode creates continuous trait variation. The majority of pairwise mutations makes an infinitely small additive contribution to the trait variations (e.g. glucose uptake by a somatic cell). Accumulation of "forward" C→T/G→A mutations lead to an increase, whereas "reverse" T→C/A→G mutations lead to a decrease in expression of the trait. Under directional selection the difference between numbers of "forward" "reverse" mutations increases.

For example, a pairwise accumulation of either "forward" T→C/A→G or "reverse" C→T/G→A mutations in a population of organisms or somatic cells would have increasing additive effect on the expression of a trait. Importantly, accumulation of T→C/A→G mutations would shift a trait variation to the left or "minus" side of the normal distribution (let's say lower glucose consumption rate of a somatic cell), whereas reverse C→T/G→A mutations would shift trait variation in opposite direction to the right or "plus" side (e. g. higher glucose consumption rate). According to this model some rare mutations if occur in a core gene may have large effect on the expression of a trait, subsequently they would be under strong negative selection [START_REF] O'connor | Extreme Polygenicity of Complex Traits Is Explained by Negative Selection[END_REF] . Whereas, the majority of reverse-complementary mutations would have rather small effect, however their effects would be additive and their accumulation under directional selection would lead to continuous variation in the expression of a trait from small to large effect depending on the number of reversecomplementary mutations or SNPs. It is tempting to speculate that the effect size of an SNP would be inversely proportional to the distance between SNP and "core" gene (Fig. 3b).

Thus, accumulation of either T→C/A→G or C→T/G→A mutations in pairwise manner over many generations will give rise to the continuous variations in the expression of a trait in a population and at the same time will maintain intra-strand DNA symmetry. Under directional selection which would favor proliferation of cells or organisms with either decreased (-) or increased (+) expression of the trait, accumulation of the reverse-complementary mutations with additive effect may change the GC content, which is one of the integral feature of genome. For example, GC content would decrease, if due to differential selection the "forward" C→T/G→A and C→A/G→T mutations accumulate at higher rate as compared to their "reverse" counterparts T→C/A→G and A→C/T→G mutations, respectively. Whereas, accumulation of the transversions C→G/G→C and T→A/A→T could modify GC and AT skews, respectively, if they occur on a chromosome in the symmetric positions with respect to the minimum or maximum of a skew.

Analysis of the patterns of common variants affecting human height, the classic example of a quantitative trait, shows that majority of 697 SNPs identified by GWAS are T*↔C (231)

and A*↔G (245) (* denotes the effect allele) [START_REF] Wood | Defining the role of common variation in the genomic and biological architecture of adult human height[END_REF] . We observe that VT*↔C ≈ VA*↔G as expected, and surprisingly no "reverse" C*↔T and G*↔A variants are perceived among other variants. minor change in the GC content, support our hypothesis that these genetic variations are additive and thus can shift the trait variation to either end of distribution (Fig. 3).

Conclusions. The golden standard of genetic studies is to examine the gene function or effect of mutation by employing isogenic strains in order to compare a wild-type strain to a strain that is genetically identical with the exception of the gene or mutation of interest. It is postulated that the use of isogenic strains rules out the effect of genetic background or differences between the genomes of organisms under investigation. Indeed, even subtle differences in distant genetic loci might influence expression of a gene or mutation studied owing to the genetic interactions. Thus, this trivial observation indicates that various forms of genetic interactions unify different parts of a genome into a whole. In the infinitesimal model an infinitely large number of genetic loci influence expression of complex trait via additive genetic interactions each of which making an infinitely small, immeasurable contribution to the phenotype. The nature of additive genetic interactions cannot be understood in a pure mechanistic manner, but rather by means of statistical approach. Mechanistic models describe specific biologically relevant functions and deal with a finite number of genes and other genetic elements, which interact in explicit manner to perform their functions, and encoded by a finite number of genetic loci. Whereas, complex traits, such as the size of cells, human height and cancer, deal with a large number of biological functions and associated with them proteins and other macromolecules and, consequently, encoded and regulated by a large number of genetic loci. In that case, interactions between genetic loci occur rather in implicit manner since they are based on the statistical relationship between their nucleotide compositions and genetic variations. Nucleotide composition of 3 rd position of a codon compensating purine bias in two first positions provide an example of implicit interaction which we call a statistical entanglement. Here, we suggest that the statistical entanglement is a contactless "mechanism" for additive genetic interactions in the Fisher's infinitesimal model. Consequently, the nucleotide composition of a gene directly affecting a trait via physical action of the protein that it encodes is statistically interwoven with many different loci in a genome, each of which making an infinitely small additive contribution to the phenotypic variations. Thus, the statistical entanglement of nucleotides in DNA challenges the neutral theory of molecular evolution. For the cellular genomes obeying CSPR there are neither neutral, nor nearly neutral mutations, because of the additive genetic interactions to which all random mutations participate in contact-free manner. In summary, intra-strand DNA symmetry equilibrium of a cellular genome, which takes origin from stochastic processes, generates regulatory, highly interconnected genetic network in living organisms and makes Darwinian evolution possible.

Methods

Analysis of mutation spectra derived from SNP variation in natural populations. NCBI dbSNP is a public collection of nucleotide variations for different species from a wide variety of sources [START_REF] Sherry | dbSNP: the NCBI database of genetic variation[END_REF]

Analysis of the nucleotide composition at different codon positions in bacteria and human

CDSs. Protein coding sequences (CDS) were selected from the curated databases, CCDS for human/mouse [START_REF] Pujar | Consensus coding sequence (CCDS) database: a standardized set of human and mouse protein-coding regions supported by expert curation[END_REF] and NCBI RefSeq for bacterial specie [START_REF] O'leary | Reference sequence (RefSeq) database at NCBI: current status, taxonomic expansion, and functional annotation[END_REF] . For bacterial genomes we exclude CDSs for hypothetical proteins. For every selected CDS we counted statistics of each single nucleotide for 1 st , 2 nd and 3 d codon positions, as well as CDS total statistics, and then compute corresponding A/T and G/C ratios, as follows we have four (A/T, G/C) ratios that includes the ratios for three different positions plus ratio for all three positions together. For human CDS we have created two dimensional plot of (A/T, G/C) for every CDS separately, in total 33420 CDS, with different colors for codon positions, and black color for CDS total. For bacteria we computed accumulative statistics, e.g. single nucleotide counts are sum of all CDSs from single bacterial species, this produces four (A/T, G/C) ratios for each species separately, in total 4038 species that contains 60417 different names of proteins.

Figure 1 .

 1 Figure 1. Effect of selective pressure on the normal distribution of a quantitative trait values in a population. (a) Stabilizing selection favors the average individuals in a population. (b) Directional selection shifts distribution of allele frequencies or genetic variations in a population to adjust trait values to environmental changes. Pink arrows denote selective pressure; blue arrow denotes directional selection.

Figure 2 .

 2 Figure 2. Nucleotide compositions at positions of a codon in bacterial and mammalian proteomes. G/C and A/T ratio for each position of a codon denoted as a red square for 1 st position; blue for 2 nd position; green for 3 rd position; and black for all three positions together. (a) Bacterial ORFs. Each point corresponds to a proteome of one bacterial strain. (b) Human ORFs. Each points corresponds to a single ORF.

Figure 3 .

 3 Figure 3. Graphical representation of the additive effect of mutations on variations in the expression of a quantitative trait. (a) normal distribution of quantitative a trait with mean value of a trait (x̄) at the centre of X-axis. (b) The normal distribution of quantitative a trait with mean value of a trait (x̄) at the extremes of X-axis. Note that the distance from core gene is plotted in additional X-axis at the bottom. Accumulation of the statistically inter-dependent

Table 1 .

 1 ). Mutation spectra inferred from single nucleotide polymorphisms in human and mice populations.

	N°	N X N	N Y N	X	Y	Total number	of SNPs	Fraction	N'X'N'	N'Y'N'	X'	Y'	Total number	of SNPs	Fraction	Difference (%)
								Human SNPs
	1 AGC	ATC	G	T	312720 0.37 GAT	GCT A	C	310244	0.36	0.79
	2 CAA CTA	A	T	227826 0.27 TAG	TTG A	T	229630	0.27	0.79
	3 ACT	ATT	C	T	2367625 2.77 AAT	AGT A	G	2375298	2.78	0.32
	4 AAA AGA	A	G	1729945 2.02 TCT	TTT C	T	1734336	2.03	0.25
	5 TAT	TTT	A	T	544308 0.64 AAA ATA A	T	543030	0.64	0.23
	6 GAG GTG	A	T	287742 0.34 CAC	CTC A	T	288408	0.34	0.23
	7 GCG	GTG	C	T	2008008 2.35 CAC	CGC A	G	2008195	2.35	0.01
	8 TAA	TGA	A	G	1448254 1.69 TCA	TTA C	T	1448196	1.69	0.00
								Mouse SNPs
	1 CGA CTA	G	T	38698 0.26 TAG	TCG A	C	38235	0.25	1.20
	2 AAG ATG	A	T	88927 0.59 CAT	CTT A	T	89884	0.59	1.06
	3 GAG GCG	A	C	50929 0.34 CGC	CTC G	T	50724	0.33	0.40
	4 CAG	CCG	A	C	56636 0.37 CGG	CTG G	T	56410	0.37	0.40
	5 GAT	GTT	A	T	63675 0.42 AAC	ATC A	T	63432	0.42	0.38
	6 ACT	ATT	C	T	384248 2.53 AAT	AGT A	G	384516	2.54	0.07
	7 TAT	TTT	A	T	103335 0.68 AAA ATA A	T	103323	0.68	0.01
	8 AAG AGG	A	G	299963 1.98 CCT	CTT C	T	299979	1.98	0.01

  Another example of the additive effect of reverse-complementary mutations is the mutational signatures in cancer, in recent version of COSMIC (the Catalogue Of Somatic Mutations In Cancer) database v3.3 -June 2022 for the single base substitutions (SBS) about 60 different types of mutational signatures are listed[START_REF] Alexandrov | The repertoire of mutational signatures in human cancer[END_REF] . Remarkably, the majority of SBS signatures in cancer 48 out of 60 can produce a change in GC content, for example in SBS1, 2, 6, 7ab, 11, 15, 19, 23, 31 and 32 the number of C→T mutations are much higher than that of reverse T→C mutations, thus accumulation of the former results in very small decrease in GC content of the cancer genomes. It should be noted that the algorithm which generates SBS signatures take in account only 6 possible base substitutions instead of 12, this operation is made by adding together pyrimidine and purine substitutions e.g. C→T and G→A mutations are presented as C→T only,

whereas C→A and G→T mutations shown as C→A only, this algorithm cannot tell us whether VC→T ≈ VG→A. Finally, accumulation of the reverse complementary T*↔C/A*↔G SNPs in human population and C→T/G→A mutations in certain common cancer types, associated with a

  . DbSNP do not track individual samples and represent all identified genetic variation aligned to reference genome. SNP/indels statistics were computed on NCBI dbSNP dataset build 142 with reference genomes: human GRCh38 build 38.1, mouse GRCm38.p2 build 38.3. Every SNP counted once from top strand, only validated SNP/indels are counted, SNPs from sex chromosomes are excluded. Every SNP/indel was enriched by left/right flanking nucleotide sequences. There are 192 directed SNP of type NXN↔NYN, where X/Y are ref/alt nucleotides. These 192 SNP's are grouped by 88 symmetric pairs based on ref/alt interchange, e.g. the count of group of AGC↔ATC was compared with the count of reverse complementary group of GAT↔GCT. In addition there are 8 self-complementary SNPs, which are presented as a single group (e.g. AGT↔ACT, CAG↔CTG …).

Acknowledgments

This work was supported by the Science Committee of the Ministry of Education and Science of the Republic of Kazakhstan (grant AP09260233) and Nazarbayev University CRP (grant 091019CRP2111) to B.T.M.; and by the French National Research Agency (grant ANR-22 CE1212-0034-01) and Electricité de France (grants RB 2020-02 and RB 2021-05) to M.K.S.

Data availability

Data sharing not applicable-no new data are generated.

Competing interests

The authors declare no competing interests.


NLRs may detect changes in their environment to accordingly modulate their signaling

What is the impact of temperature on NLR-mediated immunity?

• When isolated from NLRs, TIR domains can induce immune signaling autonomously⁶.

Thermosensitive Autoimmune NLR-mediated cell death is inhibited at 30°C

Thermotolerant

PERSPECTIVES

What about naturally occuring TIR-containing truncated NLRs?

• Cell death phenotype of 10 days-old transgenic Arabidopsis seedlings expressing TIR domains isolated from Arabidopsis RPS4 and flax L6 NLRs, or full-length autoactive L6MHV, 7 days after transfer on dexcontaining medium.

• Gene expression of major players of the salicylic acid pathway, CBP60g and ICS1², 24h after Dex-induction in 10 days-old Arabidopsis seedlings expressing pDex:L6TIR or pDex:L6MHV, at 21°C or 30°C.

L6MHV-mediated signaling is more responsive to PTI, compared to L6TIR

  des Interactions Plantes-Microbes-Environnement(LIPME), INRAE, CNRS, Université de Toulouse, F-31326 Castanet-Tolosan, France.RESULTS• Plant immunity is impacted by environmental stresses¹ , ².• Immune responses induced by nucleotide-binding Leucine-rich repeat domain (NLR) proteins are often dampened under elevated temperatures³ , ⁴ ,5 .• This is alarming in the context of climate change but the mechanisms involved are poorly understood.• EDS1 protein accumulation (A) and gene expression (B) monitored from 0 to 48h (A) or 24h (B) after Dex-induction in 10 days-old Arabidopsis seedlings expressing pDex:L6TIR or pDex:L6MHV, at 21°C or 30°C. • A. Cell death phenotype of transgenic Arabidopsis seedlings expressing pDex:L6TIR or pDex:L6MHV in different helper NLRs mutants backgrounds⁷, 7 days after transfer on Dexcontaining medium. • Cell death phenotype (A) and PTI marker genes FRK1 and NHL10 expression (B) in transgenic Arabidopsis leaves expressing pDex:L6TIR or pDex:L6MHV, 3 and 6 days (A) or 24h (B) after single or co-treatement with flg22 (PTI inducer) and/or dexamethasone. APPROACH Isolated TIR-mediated immune signaling is not affected by an elevation of temperature in contrast to full-length NLRs CONCLUSION A B Activation and signaling model for TIR-containing NLRs under temperature stress • NLRs may sense changes in their environment, such as temperature increase or PTI, and modulate the activation of downstream signaling accordingly. • Due to their simplified structure, isolated TIR domains may not sense these changes, explaining their thermotolerance and reduced response to PTI. REFERENCES: 1. Desaint, H. et al. Fight hard or die trying: when plants face pathogens under heat stress. New Phytol. 229, 712-734 (2021); 2. Kim, J. H. et al. Increasing the resilience of plant immunity to a warming climate. Nature 607, 339-344 (2022); 3. Aoun, N. et al. Quantitative Disease Resistance under Elevated Temperature: Genetic Basis of New Resistance Mechanisms to Ralstonia solanacearum. Front. Plant Sci. 8, 1387 (2017); 4. Negeri, A. et al. Characterization of temperature and light effects on the defense response phenotypes associated with the maize Rp1-D21autoactive resistance gene. BMC Plant Biol. 13, 106 (2013); 5. Heidrich, K. et al. Arabidopsis TNL-WRKY domain receptor RRS1 contributes to temperature-conditioned RPS4 auto-immunity. Front. Plant Sci. 4, (2013); 6. Bernoux, M. et al. Structural and Functional Analysis of a Plant Resistance Protein TIR Domain Reveals Interfaces for Self-Association, Signaling, and Autoregulation. Cell Host Microbe 9, 200-211 (2011). 7. Saile, S. C. et al. Two unequally redundant 'helper' immune receptor families mediate Arabidopsis thaliana intracellular 'sensor' immune receptor functions. PLoS Biol 18, e3000783 (2020).

  Isolated TIR-mediated cell death is maintained at 30°CEDS1 and the SA immune pathway are maintained under elevated temperature when induced by isolated TIR but not by full-length NLR Flax L6TIR signaling mainly depends on ADR1 helper NLR

A B ? Autoactive full- length L6 Myc Growth medium + dexamethasone

	• To investigate the impact of
	temperature on NLR immune receptors
	function and/or downstream signaling,
	we used a dexamethasone-inducible
	system to control and compare the
	activation of isolated TIR domains and
	full-length NLRs-mediated signaling, at
	different temperatures in transgenic
	Arabidopsis plants.

A B