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This paper considers a class of Caputo fractional-order genetic regulatory networks with time-varying lags in biomedicine and life sciences. Applying some features of Mittag-Leffler function and contraction mapping principle, the sufficient conditions are gained to ensure the existence and uniqueness of S-asymptotically co-periodic solution of the model. Based on comparison principle and stability theorem of linear delayed Caputo fractional-order differential equations, global asymptotical stability of the model is also investigated. The work of this article can improve and expand some existing results. [ABSTRACT FROM AUTHOR]