Quantifying landscape and flux as the driving forces for the evolution of Sex-Influenced and Sex-Linked Inheritance

Evolution is a core concept in biology. Sex-specific evolutionary pressures can lead to significant phenotypic differences between males and females, known as sexual dimorphism. At the genetic level, sex-specific selection acts on two different types of genes: sex-influenced genes and sex-linked genes. Understanding the mechanisms and consequences of sex-specific selection is crucial for elucidating the complex dynamics of species evolution and adaptation. We explored the sex-influenced inheritance and sex-linked inheritance models to study the evolutionary system of sex-specific selection, and incorporated fitness dependence on frequency selection. Frequency dependent selection has more complicate phases such as tristability while allele frequency dependent selection does not have. Utilizing the landscape and flux theory of non-equilibrium physics, we calculated the phase diagrams under the zero-noise limit, the evolution landscapes under finite noise, the barrier heights and escape times, the average flux and entropy production rate, the relaxation times of autocorrelation, and the average differences between forward and backward in time cross-correlations. The average flux as the driving force for non-equilibrium dynamics, and the entropy production as the driving force for non-equilibrium thermodynamics, provide a quantification of the system's degree of non-equilibrium, giving rise to the physical origin of bifurcation and critical transition. The relaxation time of the autocorrelation and the time irreversibility of the cross-correlations serve as early warning signals when the system approaches a critical transition or bifurcation. The lack of genotype symmetry caused by sex chromosome genes makes it possible for there to be a varied frequency distribution of sex-differentiating genes. The absence of fitness symmetry amplifies these differences. The evolutionary paths of sex-linked inheritance exhibit greater irreversibility than those of sex-influenced inheritance. It becomes more difficult to revert to the original state along the previous path which helps to preserve beneficial changes in the population. We applied non-equilibrium landscape and flux theory to the evolutionary study of sex-specific selection, revealing key physical parameters in sex-influenced inheritance and sex-linked inheritance models through quantitative analysis. This provides a new framework for understanding the dynamic behaviors and stability of systems during the evolutionary process of sex-specific selection.